How to protect your trademark in AppStore/Google Play, if they sell in the whole world?

I live in Germany. If I register a trademark here and then submit my application into the AppStore, will it be protected? The trademark will be my app's name.

I imagine that my competition could go for the same trademark in USA, thus forcing me to change my application's name?

You always register a trademark for a geographical region (smallest is one country, biggest is "the world"). The bigger this "region" is, the more it will cost you (just an example).

If you decide not to register in a certain country, someone else can do it with the same name and force you to remove your app in this countrie's app store. So will have to decide which countries are your primary market and then register the trademark there.

BTW: You will also have to choose between "word" or "logo" trademark and the registration costs multiply for each category you want it to cover (there are about 40-50).

The Madrid Protocol, an international treaty, provides a way to obtain trademark protection in multiple countries by going through your country's trademark office. You interact with the office in your native language and pay in your currency. Germany is a party to the Madrid protocol, so it should be available to you.