Emulation rules differ game by game. For most NES games, emulators are very accurate and will be lumped in with console runs since emulation neither saves nor loses time. I think (think being keyword) Nestopia is fine, but I'd check the game rules or check with a moderator just to make sure. Usually they'll have emulators that are legal and those that are not. For many more recent consoles like PS2, emulator will never be allowed because it runs/loads much faster than console. But for older games like NES/SNES era there are accepted emulators. There is a thread here http://www.speedrun.com/Speedrunning/thread/19h43 that users have been keeping up to track a list of valid emulators. You can use that for reference.

As for anything else, pretty much ask in the game forums and you'll probably get a much better answer because you'll have people who moderate/run the game there and they'll know it better than anyone else.