Do Europe laws apply in foreign (Asia) countries?

It depends on the crime and on the European nation - the EU does not have a single judicial jurisdiction yet, those powers still reside with individual nations, and thus the situation is the same for EU as it is for non-EU nations.

France for example considers any felony and some misdemeanors^* committed by, or against, its citizens to be within its own courts general jurisdiction, regardless of where the crime was committed.

Many other EU countries either consider extraterritorial jurisdiction to be within its own general jurisdiction or in specific jurisdiction for certain crimes (for example, travelling abroad to commit child sexual abuse etc).

^* The French word "crime" mentioned in the law means "felony", not "crime" (which can mean any kind of offense, from pickpocketing to first degree murder). The French word "délit" roughly means "misdemeanor", which is a serious offense but not as serious as a felony.

There is no such thing as "Europe Law". Europe is not a country nor a jurisdiction, it is a continent.

That being said, any court can decide to take any case for any reason. Usually, there are laws and/or rules which state which crimes a court is responsible for. There is no limit to that. For example, there is a law in Belgium that says that Belgian courts have jurisdiction over certain particularly egregious crimes (crimes against humanity and war crimes) regardless of where they occur, who was the perpetrator, and who was the victim. There doesn't have to be any relationship whatsoever to Belgium. Whether or not someone who was tried in a Belgian court for something which has nothing to do with Belgium actually accepts the ruling is another matter altogether. As long as you don't travel to Belgium, nothing is going to happen to you.

Now, if a Belgian court decides to try, say, George Bush or Donald Rumsfeld for war crimes committed in Iraq, and even if they sentence him to prison time, whether they can actually make him appear in court and get him to go to prison is a totally different matter. The US are not going to extradite them. (In fact, the US pressured Belgium into amending the law so that this couldn't happen, because there were actually people who asked the Belgian courts to prosecute Donald Rumsfeld.)

It all depends on the country.

Generally, it's not worth reporting the crime in Europe, in most cases. That is for two reasons.

First, penalties in Asia are much, much more severe (you definitively do not want to get in trouble with the law in Malaysia!), and while police will probably not bother investigating petty crimes, if the crime is anything substantial, they certainly will, and both treatment of the suspect by the police as well as punishment will be... let's say, harsh by our standards. No additional punishment needed, really.

Second, while it is true that some EU states (France, as an example) in principle consider their law applicable to all their citizens even abroad, this is mostly a theoretical thing which is hardly worth the paper it's written on. For most crimes, at least. Police isn't going to do anything for most crimes.

Other EU states (Germany, as an example) explicitly state early in their penal code that the law applies within their national terriorty including ships and airplanes which have the right to display their banner, followed by some exceptions in the "small print" that state which crimes are also persecuted abroad.

Those exceptions are high treason and terrorism, crimes against humanity or life (read as: torture, murder, trafficing, forced prostitution, trading organs, etc), kidnapping, and crimes involving serious bodily harm, bribing officials and delegates, counterfeiting, piracy (in terms of capturing ships, not software!), shooting at airplanes, sabotaging nuclear plants, that kind of thing.

So... unless you are really one tough bastard of a supervillain, this doesn't apply to your usual everyday crime.

You can, however, provided you have sufficient evidence to avoid risking a sentence for false accusation, do it nevertheless. It costs you nothing, and it remains in the records. Everything and anything you report remains in the records forever.
It has no legal consequence whatsoever, except when the next thing happens maybe in 5 or 6 years, possibly to someone else, the policeman looking up the name will raise an eyebrow (I've just had this happen last week). Which may make the difference between the policeman judging your situation as "plausible" or "not plausible".

First of all, I concur with the accepted answer that it's generally best to report in the jurisdiction where it happened. There are of course exceptions, for example, when it comes to small crimes that have severe punishments in the occurring jurisdiction. In that case you might want to hold off on reporting or only report them when back home.

On the other hand, I do see the benefit of reporting them in Europe too, though there are things to consider. One of the reasons you might want to report them in Europe (as well) is that it doesn't hurt to have an extra set of eyes, they might have more resources (e.g. in case its one of their citizens they may have a criminal background that the occurring jurisdiction didn't enquire about or was denied access to) or they might take the issue more seriously.

Some of the things to consider before reporting outside the occurring jurisdiction:

• Evidence: if you have any evidence, that might help the other jurisdiction get started. If they have nothing to go on, it's less likely that they are able to produce leads.

• Severity: a severe accusation is more likely to be considered. Your pen getting stolen, is not as important as someone who got murdered or raped. The latter examples are more likely to be considered, especially because it could be damaging

• Connection: the crime should also be connected to the reporting jurisdiction. For example, if it's a Dutch citizen who does something bad abroad, it might be helpful to report them to Dutch authorities as well because the offender may have a history which can help the investigators locally.

In addition to the above, you might also want to consider specific organisations that are aimed at preventing specific crimes. Especially when it comes to the European-Asia connection, a popular method is for European sex offenders to go to Asia because they easily acquire a position of power (because they tend to be a lot richer than the locals) and there are few controls on working with children (whereas most European countries require background checks).

One organisation dealing with this (children's well-being) specifically is Terre Des Hommes, which supports some of my previous points in their description of child tourism in Cambodia:

Children in lands like Cambodja are specially vulnerable to the risks of sex tourism. Due to the poverty, rich tourists seem like an appealing option to many. Children and families are often not aware of the dangers and the police does not have the man power to combat the problem.

The organisation's Wikipedia page lists (and links to) some more examples of local and international cooperation that actually produces results.