After Barret says "C'mon, newcomer" at the very beginning. I press my key for O which is fine. But then I get into the first battle with the Shinra guards, none of the keys are working. Is this caused by something on my end or Bizhawk's end?

99.9% of people in the site don't even run the game. Tell this in the game's forum please.

Bizhawk has some weird hotkeys, maybe you accidentaly pressed one which paused the game or something? I would suggest removing all hotkeys (once in the hotkey window just click on "Misc" at the bottom and theres a option for doing that) and re-adding the ones you actually need