Is it possible for us to move our medals around, and pick which 4 appear in our medal cases on our profile page? It would be a cool little feature, and I'm sure several people would like to show off their various achievements rather than having the site pick which ones go on their main trophy case.

It's been considered, yes. I think ideally, people would like to have control over everything on their profiles.

I think you should be able to control them as well.

allowing this would seem logical

This sounds like a good idea, too bad I don't attend tournaments.

But if I did, I'd definitely want this to be a option.

I think this was labeled as one of those ideas that was too hard to implement for the amount of convenience it would give. Sorry, I don't think it'll happen any time soon. (if ever at all)