I don't live in Europe, but was just wondering if I should register these? Do people in europe usually go to these instead of .com?
Why are you wanting to register .co.uk & .eu names for? Are you planning to target those audiences? If you're wanting more of a global scale, go for the .com because as they say, .com is king
It's just that I'm kind of paranoid that if my site gets kind of big, people will steal the name and then make their own site with it. I don't know if that's legal though.
The answer is no. The .co.uk extension is used only by those from the UK and although the .eu extension is becoming increasingly more popular with European companies, it still is nowhere near as popular as .com. From a European perspective: if you have a pan-European company then .com is king and .eu is the king's wife (certainly not .net). If you have a company in a European country with most of it's business there, then go for a national extension. BTW, you need to be a citizen of one of the EU member states to be able to register a .eu domain.
These are highly recognized and respected in their respective countries. If you wish to establish a presence in these areas they are definitely a great investment. If your looking to establish on a global scale the dot com is the way to go.