You ask a strange question ?
Can you program a game? If not, I'd suggest taking that on first.
If you can program a game, you want a game for many users with a server? I have no idea about how feasible that is (I don't work in the game industry at all), but others may know that. Anyway, aside from requirements from FB (which I don't know but no doubt FB can tell you that), there are mostly 2 reasons for having a server. First reason is to store data that you cannot or want not to store in the app itself. The server than acts as an off-app storage that you control. The second reason is if you want users to exchange data (eg scores, play time and what not) or “meet”, eg game lobbies where players wait until everybody is online to start a game.
As for publishers, did you try a search engine? I mean, unless FB explicitly forbids publishers to tell to the world they made a FB game, I don't see a reason why they would even try to hide it. As such they should be findable in that case with a good search query.