Good thinking, and regarding Euro institutions there's lot of regional differences- for instance, brits have steps to take which are pretty set in stone with the Centres for Doctoral Training; basically it's the opposite of the process from the prior comment, it's the system that allocates someone for you. Denmark is free if you're native or from EU- Finland however is free for all (according to peers and the net) and would gauge they're both tad more flexible than UK in terms of requirements. Anyway, there's plenty options!
Regarding the selection process, although they're a clear benefit perfect grades aren't a must and it's alright to be from outside of a top ranking school- but you'll have to bring something to the table to make up for it. Whether that is a solid paper, practical achievements along recommendations or whatever again depends on the place. I think there's just so many ways to show something that can be impressive without knowledge about the latest thing or dropping walls of equations for the sake of it haha.
Best of luck!