I'm sorry to bother, but I wanted to know if there's any rule against this game being ran in a 3ds, if there is could I use an emulator like Desmume and if so would I need to show any proof that I would not be using any kind of cheats? But if there's not would there be any need of a capture card to submit a run or could I just capture the video with a camera for instances my phone. That's all I want to know for now, thanks in advance.

There isn't any rule against the game being played on a 3DS but you could use Desmume as long as you don't use cheats. You can record with a camera or with you phone as long as you time it or time it after the run correctly