<Snipped quote by Stale Pizza>
Now, I'm not Rune, so I can't really answer your question about whether you'd be able to play as a Gym Leader but not a trainer… but would you really want to? I mean, Gym Leaders don't tend to do all that much, because they've got their own battles to take care of, and we might not even get to your Gym for ages. xD
But at the same time, Gym Leaders are not static and flat characters. In the anime and the game, it's frequent that you see the Gym Leaders wander around before you even get to the Gym, and a few even become temporary companions, either for the whole series or a part of it.
Also, there isn't really any rules to naming Gyms, but generally I like making it connect to their main Gym type, either directly or indirectly.