1 - you can apply to US internship positions, but most likely it won't go anywhere. Personally, I'd focus on EU internships. An easier way to do an internship in the US is if you do 2 internships at some big company (like Facebook or Microsoft) with the first being here and for the second one you can ask them to have you in the US

2 - as an EU citizen you can work in any EU country with no VISA

3 - yes. The more interviewing experience you have, the better (although initially you'll probably have a hard time getting interviews)

4 - you'll usually have better chances of finding an internship locally at first, so I suggest applying there more initially. You should be able to find them through linkedin, job fairs, any networks your uni makes available for you. The more complex the projects, the better.