If you do an American license you need an American medical.
If you do a European license you need a(n) European medical.
Whichever country's license you do, you need that country's medical. It doesn't matter where you train, it's which license you get that decides which medical you need.
Once you have the license, it allows you to fly that country's aircraft. You (generally) can't fly a European aircraft on an American license and vice versa. If you want to come back to Europe to work after training in the states, you'll probably have to convert the license. Depending on how much experience you have this can take 2-6 months and cost a few thousand pounds.
School recommendations? There are a lot of good schools out there and most of us only have real experience of one or two. It would be easier to find out the bad schools and avoid them. After that find schools which suit your requirement and then see how well set up they are.
Good luck.