It's basically formed from any 'other-worlder' trainer who beats the Elite Four. It's kind of a mini-league? I don't know every detail, but when Hermana was brought in, she was told she could find any location within Johto, Kanto, and Sevii to build her gym.

Probably...there's going to be rules for any that are in the same city as another gym, and since normal gyms have type themes, you'd need a specific theme as well...but that's the most I can think of.