Is it rude to speak Swedish in Norway?

It is neither rude, nor arrogant, but I am not sure if I see any point in doing so. If you don't understand Norwegian very well, you are very likely to have larger difficulties with the rural dialects in Trøndelag, and especially if you have a foreign accent when speaking Swedish, you can not take it for granted that all Norwegians will understand your language with ease.

You are right that Norwegians and Swedes are often communicating in their own respective native language. As you have noticed yourself, it can however be difficult to understand Norwegian even if you are quite good in Swedish as a foreign language. The same applies to native speakers as well. Even if the languages are similar, they are IMHO different enough to be barely mutually intelligible if you have absolutely no practice or experience with the other language at all.

Research has shown that mutually understanding of the Nordic languages (this includes Danish as well) has declined among younger people, living in border proximity is a great advantage and Norwegians generally understand Swedish better than the other way around. The decline in proficiency among younger people is often attributed to the spread of cable tv and a broader selection of domestic tv channels. Commercial television didn't emerge in the Nordic countries until the late 80ies and before that, each country had only one (Norway and Denmark) or two (Sweden) public TV channels. The only option for many people for some variety was, at least along the borders, to watch tv from the neighbour country.

As a Norwegian I will say that we mostly wouldn't find it rude at all (rightfully so), problem is that the response will be in Norwegian and that might pose a problem for you as you have stated. For interacting with anyone within 15-50 years I would say just speak English. For older people it might be an idea to ask if they speak English first. Also keep in mind that the dialect Trøndersk in Trøndelag is potentially problematic to understand even for some Norwegians (that are too lazy to try properly IMHO).

As a native Swedish speaker I will talk Swedish when with Norwegians and Norwegians with Swedes will usually speak Norwegian. However depending on your how comfortable you are with the respective language, you can throw in the Norwegian words you know and try to speak with a more similar melody/prosody.

Limit your vocabulary to a more basic one for less confusion. Still there are many words that are completely different, so be prepared to explain them in Swedish or English. (One thing that comes to mind is sidewalk, trottoar vs fortauet).

The only thing that would be rude is if you cannot understand them, or they you, and you insist on speaking Swedish even though you could switch to English.

This answer is not specific to Norway, to which I have never been, so hopefully I am not missing something important here.

But try learning a simple phrase in a local language (in your case you may already know how to say that:

Hello (excuse me), do you speak English / Swedish / etc?

In my experience it is perceived in a very positive way by locals (way better than asking the same way in English or any other non-native language). Usually people asked that way are willing to help a lost stranger so much that many of them really go out of their way.

Besides that, any in any Western country the idea of tolerance is quite big nowadays, so you probably will not have any problems whichever language you speak -- although it's definitely good to try your best when talking to locals.

My answer will be a bit more general and won't limit to Norway/Swedish only.

It may or may not be rude, depending on what you want to achieve and how you approach this.

First let's point out one thing. The goal of any communication is to efficiently exchange messages between participants. Whatever method works, if you've managed to pass the message your communication was successful. Having this in mind it's absolutely fine to use any means of communication you have at hand, especially finding any mutual language is absolutely OK as long as both/all parties agree to use it.

If I were in your position, I would start with Norwegian, just to show courtesy. Then once I felt no longer comfortable (e.g. I felt I am unlikely to understand an answer to my question) I would ask if we can switch to some other language, suggesting English or Swedish. Sometimes (if I'm sure the other party knows English, e.g. at work or at a hotel) I would just switch to English apologising that my Norwegian reaches only that far. In Norway and other countries with a well developed English learning system it is very likely you'll never get to the point where you can suggest changing the language. It'll be the local person who'll immediately recognise you have difficulty speaking their language and offer you a language switch, probably suggesting English. There you are, continue in English and if there are some problems with that as well, you can suggest you know also Swedish (and whatever other languages, especially somehow related to the local language, so German/Dutch are good options too) if that can help the other participant.

Last few resorts, when finding a common working language fails is to:

• use a mix of languages, ask for clarifications whenever needed
• add hands to your conversation; it's amazing how much can you explain non-verbally when in a real need
• use Google Translate (as you could see in Russia during recent Football World Cup); it can really work and does more than decent work, especially if one of the parties can use proper English
• find an interpreter

As for using Swedish rather than Norwegian to show courtesy only when you know your communication is going to be better in English, I would be very cautious. You aren't actually using the local language anyway so you might be involuntarily suggesting you don't care about the difference (so actually the local language). While it should no longer play its role, remember the history between Norway and Sweden wasn't always as good as it is now. Norway gained independence from Sweden only in 1905. So I would not go that path.

Few examples from my own experience, some related to Norway specifically, some in other setups, that might give a better understanding.

1. I was trice in Norway. When I was 6, 13 and 26. As a kid I knew neither English nor Norwegian. I didn't have to communicate in any of those languages in general as I have Polish speaking family there but I remember I was happy to learn at least few words from my aunt's neighbour. When I was 13 I already knew very basic English. I was able to communicate with my Norwegian colleague (about the same age) using a mix of English, Polish, Norwegian, hand-waving and pointing things. We understood each other perfectly. I also had a funny encounter with a local who happened to ask me something in the train station. I managed to say (with a mistake) the only sentence I knew: "I for stor ikke" (should be "Jeg for stor ikke"), which means "I don't understand". The guy thought it over for a brief moment and immediately switched to English (and I was so happy to understand him and be able to answer that I remember it to this day ;-) ). During my last trip I was using English entirely and it didn't fail me once.
2. I travel a bit and usually I learn at least basic greetings and things like "Thank you", "Yes", "No" etc. Sometimes I push a bit more, and for example in Lithuania I was able to order food in Lithuanian. Yet usually that's all I can say in a local language and switching to English the way I described worked just fine with just few notable exceptions I'll mention later
3. I've been several times in Czech Republic and Slovakia, two countries which were separated even more recently than Norway and Sweden. These were mostly business meetings and most of the discussion was done in English. Yet sometimes we were using a mix of English/Polish/Czech/Slovakian (depending who was in a meeting) as those Slavic languages are so close to each other we could mostly understand each other. Only later I found out that whenever the guys spoke between themselves they always used their native languages so Czechs were speaking Czech and Slovaks - Slovak. It was common for them. Yet I believe if I tried to use poor Slovak against Czech and especially poor Czech against Slovak it wouldn't be too welcome, especially when I am able to communicate in English.
4. In Lithuania the knowledge of Polish is common. Yet some younger people reject using it due to some bad history between our countries. Usually English worked just fine. Yet I had a situation where I had a problem as no-one could speak even at mediocre level (at the airport!) and it turned out the seller was perfectly happy to speak Polish (by sheer accident, I just referred in Polish to kids accompanying me and the seller happily said in Polish "Oh, I can speak Polish"). Yet I know enforcing Polish would be plain rude unless clearly accepted/offered by the other person.
5. In Italy English failed miserably. In several situations I was able to say more in Italian (and I really don't know that language) than the guys could understand in English. At one case I've asked as politely as I could "Parli inglese" (Do you speak English) and the guy whom I asked just jumped into the car and drove away (he just stopped a moment before at the fuel station so I don't think he was going to depart that soon).

You describe your first option as "Norwegian with Swedish mixed in". Given the language proficiency you describe, I think that a better description of your options would be to speak plain Swedish or to speak Swedish with a few Norwegian words mixed in. What I mean is that, since you know Swedish much better than Norwegian, what you say will likely be interpreted as Swedish either way. The few specifically Norwegian words that you may know and use will have much less weight when they subconsciously determine which language you are speaking than the specifically Swedish words and expression you cannot avoid using. On top of this your pronunciation/accent is likely to sound more Swedish than Norwegian no matter how hard you try. The thing is that a Swedish person will probably notice the Norwegian flavor that you may add to your Swedish, but to a Norwegian the Swedish flavor will still dominate their impression. People pay more attention to the things that doesn't fit with the language that they know well and speak themselves.

As a native Swede living in Sweden, I know several Norwegians who have lived in Sweden for many years. I know that they have adapted their vocabulary and and pronunciation quite a lot and that they would probably consider themselves to be speaking Swedish. Still I know that I and other Swedes instinctively perceive them as speaking Norwegian. I also know Swedish people who have lived in Norway for a long time and experienced the reversed situation.

That being said, I would recommend mixing in Norwegian words and expressions as you learn them. It will make it slightly easier for them to understand you and it will make you more comfortable speaking with Norwegians in the long run. Either way they are likely to think of you as speaking Swedish, but unlikely to find it rude. If someone would actually find it rude, it's unlikely that there is anything you could have done differently to avoid it.

Go with your limited Norwegian to start (even if extremely limited). Just say enough to say hello and to explore what language to speak best in. Personally, I think you will have best experience if you just struggle by in Norwegian especially if the conversation is short (buying things, restaurant, taxi, etc.) It's just that you are more in the culture and the country then.

If you need to have a long substantive conversation (e.g. solving a business or technical problem for instance), of course find the language of greatest mutual intelligibility, likely English.

My take on this as a Norwegian.

Speaking Swedish would not be seen as rude.

However it's fairly pointless. With the exception of people interacting with swedes daily we will understand English better than Swedish. - If a Norwegian and a Swede converse we will switch to English if we have difficulty understanding each others. This goes double if Swedish is not your native tongue.

It seems that all Nordic people understand Swedish, but the Swedes generally don't understand the other languages.

In a professional setting Swedes would speak Swedish with Norwegians as it's not too hard for Swedes to understand Norwegian. When speaking to Danish people English would be required, unless you live in the south and is used to their dialect. The same goes for Icelandic, but you might be able to communicate with them in Swedish if you live close to the Finnish border as the Finnish-Swedish dialect is similar to Icelandic.