I would make this a comment if I had the reputation, but:

As a late-twenties working Norwegian living in Helsinki I can confirm that English is more useful than Swedish in Finland. The only people I've spoken Swedish to are those who want to practice their Swedish. On the other hand, being able to understand all official communications and signs is useful.

The situation might be different in primarily Swedish-speaking areas.

English alone will allow you to speak to (just about) a majority of adults in your all your different groups of countries. Adding German alone achieves a comfortable majority in all your groupings of countries

Let's put some numbers on this. We can look at the numbers who speak various languages in each country as listed on wikipedia which considers only those aged 15+ (reasonable for these purposes). Note that these numbers date from 2012 and the number of English speakers now may well be higher given English is generally more prevalent among younger Europeans than older ones. Similar numbers don't appear to be easily available for the Swiss (or Norwegians or Icelanders), but if look at native languages we can get put a lower bound on the number speaking each language.

The first group of countries you list (Belgium, the Netherlands, Luxembourg, Germany, Switzerland, Austria, and Denmark) has a combined population aged 15+ pf about 105 million, of whom about 61 million speak English - so English alone will allow you to speak with the majority (58%) of adults in these countries.

If we then add the second group of countries (Norway, Sweden, Finland and Iceland) the total 15+ population increases to about 122 million. Adding the number of Swedes, Finns, Norwegians and Icelanders who speak English (last two at around 90%) we have around 74.5 million speaking English. So again English alone will be enough to speak to a majority (61%) of people.

Finally, if we add all of France and Italy, we have a total population aged 15+ of about 222 million of whom about 111 million speak English, i.e. around 50%. However, adding any one of German, French or Italian would be sufficient to make this a comfortable majority.

Specifically if you add German (and conservatively assume that all German speakers outside of Germany, Austria and Luxembourg also speak English, whilst all English speakers in those countries speak German) you reach about 90 million (85%) in group 1, 103 million (84%) in group 1+2 and 140 million (63%) in group 1+2+3.

Doing the same for French, we get 64 million (61%) in group 1, 78 million (64%) in group 1+2 and 142 million (64%) in group 1+2+3.

Finally for Italians the numbers are unchanged for groups 1 and 1+2, whilst for groups 1+2+3 we get 144 million (65%).

German would get you quite far: This map on the Wikipedia page of the German language shows how many people understand german:

English will get you understood by the vast majority of people in the Scandinavian countries (including Finland and I believe Iceland) and I at least by the vast majority of the younger generations in the Netherlands, Germany, Austria, German Switzerland and probably Flemish Belgium.

French will top up that list by getting you understood in France, French Belgium, French Switzerland and Luxembourg.

Aside from these two, the most helpful language in getting understood across multiple European countries is probably Russian which most of the older generations in the former Soviet Block states learned in school. In addition, there is a non-negligible similarity among the different Slavic languages that might facilitate understanding between non-Russian Slavic speakers and Russian speakers – like the speakers of various Romance languages (most notably Spanish and Italian) can often understand each other when speaking slowly, clearly and in basic sentences.

In my experience, the southern European countries don’t really have any commonly understood language other than the respective native languages (Italian, French, Portugese, Greek, various languages of the Balkan); in tourist-heavy areas, a significant tourist language may well be understood by employees in the tourism sector (i.e. you can probably order your food in German on Lake Garda). The similarities and limited intelligibility between Italian and Spanish are known, but especially French – despite being geographically in between them – is notably different so French can’t be used as well in those countries and vice-versa.

As for Scandinavia, the Scandinavian languages (except maybe Icelandic; I don’t know enough about that) are similar enough to claim mutual intelligibility. In Finland (Finnish is not a Scandinavian, Germanic or even Indo-European language), Swedish is taught at all public schools and Finlandswedes are a relevant minority but native Finnish speakers tend to have a strong dislike of speaking Swedish at all and often forget all they learnt in a couple of years (unless they are employed by the government or a company that requires knowledge of Swedish such as the state railways). Thus, you could, if you wanted, add e.g. Swedish to the list of languages as it will get you understood across most of Scandinavia. However, almost everybody has also learned and still has a decent grasp of English so there is almost no need to add Swedish to your list when English can do the exact same job but also cover the UK and the Republic of Ireland in addition to serving as a backup language in other places.

I don't think you will get much improvement over English by studying more languages, it follows the law of diminishing returns. However, consider these maps which may be useful in deciding which ones would be most useful:

Source: Map of the most spoken foreign languages in the EU by country

Disclaimer:

I am Norwegian by birth, but grew up in Spain; so I basically have a fluent level of Spanish, English, Norwegian and Catalan. My intuition for languages is quite good.

Storytime:

I travel quite a bit, so I am often in airports. I tend to speak with people of different cultures, and once in the airport of Oslo I met a guy that did not speak English, Norwegian, or Spanish.

He did speak German and French, though. We spent hours talking, by swapping from one language to the other. Body language is universal, anywhere.

Point 2:

Personally, I can understand Dutch with ease, yet I don't speak it at all. I spent a lot of time at a friend's place from the Netherlands during my adolescence, and started picking up phrases and words just from listening to him talking with his family. Now I can basically understand any Dutch which is thrown at me, and by "understanding" Dutch, my ability to understand German is heightened.

I don't believe the average person from Norway would just understand Dutch, and much less German, though.

Point 3

As a Norwegian person, Swedish and Danish are basically dialects. There are Norwegian dialects which differ more from "standard" Norwegian than these languages. Finnish is much different, it has basically nothing in common.

Lastly

Have a look at this article (it helps if you understand Swedish, lol):

Håller språket ihop Norden?, by Delsing & Åkesson, explaining how Norwegian people are better at understanding Swedish and Danish than they are at understanding us. There are historical, linguistical and social reasons for this.

Similarly, this video explains some of it: video

Europeans who speak one at least foreign language, generally speak English. Hence the language(s) you're looking for are those spoken by large number of people that don't speak a second language.

That makes the list rather obvious. The obvious #1 is German for Germany, Austria and parts of Switzerland. Sure, there are Danes who speak German, but those are likely tri-lingual (Danish, English, German). French is useful by the same logic in France, Belgium and again parts of Switzerland. The other two large languages are Italian and Spanish, but those are only really useful in their respective countries.

Smaller languages like Dutch are essentially pointless. Because they're such small languages, a large part of the country will speak at least rudimentary English. You're not going to find many Dutch who do speak German, but not English.

Focusing on the countries from your first sentence (Belgium, Netherlands, Luxembourg, Germany, Switzerland, Austria, and Denmark) I say that out of all those languages spoken there, German would be the most useful. It's spoken in Germany (duh), Austria, and at least parts of Switzerland, Belgium and Luxembourg. In the Netherlands and Denmark it's not totally uncommon to learn German at school, however most of the time I've met with people from the Netherlands I've conversed in english with them as they felt more comfortable in that language than they did in German (and I don't speak any Dutch at all, so that was out of the question).

Including Norway and Sweden (second part of your question), you might also get the furthest with German. Again, it's not totally unheard of to learn German at school in those countries, but the final result may vary. When I traveled Norway there was one camp ground owner who conversed really well in German, everyone else preferred to speak English.

Out of the scandinavian languages Norwegian is the most useful. It enables you to understand spoken Swedish to a degree (tests have shown that a Norwegian speaker can understand about 80% of spoken Swedish) and you might be able to read simple texts as well, if you can get your head around the different spelling of some words. You'll also be able to read Danish, which comes of no surprise since Norwegian bokmål derives heavily from Danish. Understanding spoken Danish is a very different topic, though. The pronounciation is very different from Norwegian so it would be really hard to follow along.

Also, knowing Norwegian also makes it possible to at least get the gist of simple texts written in Dutch or German, however I don't think you'd be able to understand the spoken languages.

Icelandic is a Germanic language and is closest related to Norwegian (and Faroese), so that brings us back to Norwegian as the most useful language. :)

Adding Finland makes it a lot more difficult, though. Finnish is not a Germanic but a Finno-Ugric language and bears little to no resemblance to the Germanic language tree. While there might be a few people who speak Swedish, Norwegian or other languages, I'd say English is by far the most useful language in Finland apart from Finnish.

To bring this post to a conclusion: based on my personal experiences I say that apart from English the most useful languages in the countries you listed would be German and Norwegian.

As already stated and backed up, Basic English is the language that provide you the highest chance you will be understood by th natives.

There is also one universal language that helps anyone to cover the rest - the ol'good hand-foot language.

I has no grammar, no vocabulary. It is solely powered by participants' creativity and will to understand each other.

Almost any foreign language may cause problems in post-soviet regions when speaking with older people - they were forced to learn russian and punished for learning any other (even polish in Czechoslovakia). many of people hated russian by heart and the rest forgot the language because after the fall of iron curtain the russian became obsolete. Every year since 1989 this is getting better and better and more and more people are learning some foreign language.

Other people have given you the data about which languages have the most potential.

Let me address the generalized question of "what language should I learn" with an anecdote from Surely you're Joking, Mr. Feynman! He decided to study Spanish because it was more widely spoken in South America where he wanted to visit and ended up on an academic exchange in Brazil (chapter "O Americano, Outra Vez!") where they speak Portuguese instead.

Don't get too worked up about studying the right language first. Study the language that you are motivated to learn for whatever whimsical reason, even if it has a small population base. The more languages you learn, the easier it is to learn languages. A lot of Dutch working in South America pick it up faster than the rest of us (say 6 months) because it's their 5th language and no big deal by then.

More important is your immediate access to language materials. If you've got a native speaker available to you, that's the one to learn. I had no great need or desire to learn Bulgarian (5m speakers), but there are 2 people in the office from there, so I got the phrasebook and found that it's kind of a cool language, too.

The other way is to choose the language based on the programs you like to watch. Walter Presents on Channel 4 has a wide range of good non-English series to give you a quick intro.

It is a boring answer but, once you know English, there is no obvious next language to learn. For a specific country, it will be useful to learn the local language but for general use, it is hard to find another.

In general, when in a country where English is not the primary language, I open conversations with: do you speak English? If possible, I learn how to say this in the local language. In some countries, I have stopped doing this as people seem offended by my doubt.

Further south, French will have some value. Obviously in France but I have found it useful in Portugal where there are people who know French but not English. However, knowledge of English is growing rapidly and hence the value of French is dropping.

Be careful of using French in Belgium as a Flemish speaker might prefer you to use English.

I have frequently heard Danes and Swedes using English to speak to each other rather than their own languages. Most know English so well that this is the easier route. I have even heard French tourists in Italy use English which I found more surprising.

Are you a native speaker of English? My wife and I have often noticed that people are more willing to speak English with her than me. The explanation seems to be that I look and sound like a native speaker but she does not. So, with her, English is seen as a neutral third language. This suggests a non-obvious strategy: learn to speak English worse.

From my experience French and German. English is widely spoken in the Scandinavian countries and the Netherlands. German is great in German speaking countries, but also eastern Europe (certainly Hungary, where you can find people who speak German, but not English). A lot of Italians have worked in Germany and speak the language from my experience. Speakers of Romance languages (Spain, Italy) find French easier to learn, though of course some of the Costas have a lot of English living there and thus English is widely known in those parts.

When you add to the mix France and Italy, you're pretty much out of luck. The French speaks French, and Italians speak Italian. Most of both can deal with English, and you may well find Italians who speak French or German, especially in the North that is closer to the borders, but don't count at all on speaking German anywhere in Italy.

Don't expect people to understand esperanto or interlingua, at least in Italy.

Then again, you may find anything! I remember when I was in Wien and I, a basic German speaker, tried to prepare the "perfect" German sentence to ask for underground directions: I stood at the underground exit, I aimed for a passenger to come by, and then, with all the confidence I could be capable of: "Entschuldigung, ist dieser der richt..." only to be be cut off: "I'm sorry, I can't speak German!".

German would probably be a good choice.

With English as a baseline, you already have a very reasonable coverage all over Europe. That is, not everybody will speak with an accent that is bearable without causing convulsions, not everybody will be willing to speak it or attempting to understand (French, in particular), and not everybody will be able to follow and understand a political discussion or a scientific debate (but not everyone will be able to do that in his native language either). Overall, it will "kinda work" for a good share of the population, if general understanding is the goal.
In theory, coverage among the younger generations (under 50) should be something like 95-99% because there is no way you can grow up and not go to school, and there is no way you can go to school without having English as mandatory language for at least two years or so in the overwhelming majority (possibly all, but I won't claim that!) of west European countries. So unless you are a migrant coming from some place where no English is spoken and where no schools exist, it's pretty much guaranteed that you have at least basic coverage.

Now, about German and Benelux as well as East France (which is really kinda West Germany) and northern Italy (which is definitively Southern Germany) you are good to go with German. Same for Austria, and a good part of Switzerland.
In fact, in some parts of Northern Italy, you had better not speak Italian! A very significant amount of people there speaks Italian only because they must, and they pay as little tax as they possibly can to cursed Rome. They're more German than the Germans, and they hate Italian with a passion. I'm acquainted with someone who is now "retired", but he used to be in what's the petty north-Italian equivalent of ETA or IRA, whatever you call it, when I was a teenager. A certain way of having him punch you in the face even today is to point out that technically, he is Italian. I don't think they still do bombings and puncture tires nowadays, but sure enough if you disqualify yourself as Italian in the wrong region, you get a frown at the very least, also the worst possible service, minimal courtesy and cooperation, and you're going to get cheated (and spit in your food), if there is any chance of doing so. So, consequentially... no worries about understanding German there.
It's surprising how well people understand and speak German in some parts of Spain, too. You wonder why. Must be due to tourism, I guess.

Scandinavian countries (not counting Finland, their language is abysmal, but luckily they speak English), I wouldn't know about their side to be honest, but I found it embarrassingly trivial to understand at least some, and to learn enough Swedish to actually work there and get around, all over a period of just two weeks. Try and learn French in two weeks, heh.
It's sooooooo similar. So the assumption would be that if you do not plan to do a scientific discussion, you are probably good to go with German, too. Also, they're excellent at English, and since that was a baseline assumption, you're good to go.

France may be a real problem, not because they couldn't understand or speak English or German, but because they won't do it. I've personally never had this problem since I've had French as first foreign language, and don't bother speaking it the same as if I'd speak German or English. But I've heard soooo many times from people that you get zero cooperation otherwise. Guess how surprised I was to find out after 12 years (!) that a French friend of mine actually speaks German without an accent. Only just, he'd rather die than do so.

English + 1 to be understood

I presume that you want to learn a single language to maximize your reach, therefore my suggestion will be different to the other answers.

I'm referring to Belgium, the Netherlands, Luxembourg, Germany, Switzerland, Austria, and Denmark.

English is a safe bet in all these countries (personal experience in Austria and Germany). German increases your chances of things going smoothly, but my recommendation is to keep your "slot," in these countries it is easy to find a passer-by who will interpret for you if need be.

Will it be possible to add Norway, Sweden, and possibly even Finland and Iceland?

English is once again a safe bet; in Finland I encountered language issues but once during my 4 months there. Scandinavian movie theaters tend to air movies in the original language with subtitles, people pick up English from a young age,

And how about the northern parts of France and Italy?

These are the harder ones. In my experience you need help to understand the Italians once you go past tourist hot spots, but they try and the passers-by will be happy to help.
In France you need to observe the language and cultural barriers, I have anecdotal evidence of them being rude (e.g. receptionist not handing over customer's key) in response to not speaking the language and starting a conversation on the wrong foot. Once they see your attempt at speaking French they will likely ease up and might even offer to switch to English.

Interlingua (!)

From Wikipedia:

Interlingua is an Italic international auxiliary language (IAL), developed between 1937 and 1951 by the International Auxiliary Language Association (IALA).

Interlingua's greatest advantage is that it is the most widely understood international auxiliary language by virtue of its naturalistic (as opposed to schematic) grammar and vocabulary, allowing those familiar with a Romance language, and educated speakers of English, to read and understand it without prior study.

It ranks among the top most widely used IALs, and is the most widely used naturalistic IAL: in other words, its vocabulary, grammar and other characteristics are derived from natural languages, rather than being centrally planned. Interlingua was developed to combine a simple, mostly regular grammar with a vocabulary common to the widest possible range of western European languages, making it unusually easy to learn, at least for those whose native languages were sources of Interlingua's vocabulary and grammar. Conversely, it is used as a rapid introduction to many natural languages.

Interlingua literature maintains that (written) Interlingua is comprehensible to the hundreds of millions of people who speak Romance languages