While it wasn't marked as explicitly banned it seems to be implied with the "most others" but I was wondering if Mesen 0.9.9 is acceptable? I have seen in various reports it is rather accurate for an NES emulator.

Yes, Mesen is quite accurate emulator, so it's allowed. It's in the rules as well : "FCEUX, Nestopia,Mesen and Bizhawk/NEShawk are acceptable emulators" .

If you're playing on an emulator and you don't have a controller, would a keyboard be allowed? Would that be a different category?

Keyboard is allowed in the same category.