i do not have a capture card, but couldn't find any concise rules on whether or not emulation was allowed.

if they are, what emulators are allowed?

Yes, emulators are allowed. I'd say the more reliable ones are FCEUX, Mesen and Nestopia, so if you can get one of those, you're good to go. :)

Quick question: Are those still slower than actual console because of FPS, or are they equally fast?

Nope, they emulate NES consoles very accurately, so there's no difference in terms of speed.