Is there an arcade in the hotel? I'm looking for a specific arcade game and this resort may have it.

There is. It's called The Game Station, it's on the 4th floor.

What is the specific game?

Not anymore. The arcade that used to be on the 4th floor is now closed. That location is now a childcare drop off center for kids called "The Pixar Play Zone." https://disneyparks.disney.go.com/blog/2018/03/reservations-now-open-for-pixar-play-zone-at-disneys-contemporary-resort/

Yes. There is an arcade on the fourth floor behind the gift shops. I do not recall seeing F-Zero AX, but I may have just missed it.

I don't recall

Specifically it is called F-zero AX. Its a racing cabinet.

Yes, there is one next to the gift shop on the same level as Chef Mickey's.