Definitely definitely yes to banning illegal or impossible to own pokemon. Not only that, but seeing nature/ability is a must these days.
Not only GTS, but I think Wi-Fi in general needs more work. Like, why do all double battles need to have only 4 pokemon? Maybe, just maybe I want to have an actual battle and use all of my pokemon? I'd just like more customization. Maybe have the whole someone creates a lobby, someone joins thing. That way the host can customize the match (how many pokemon used, single double or triple, items or no items, anything) and the other person can decide if they want to play with them or not.