Is it OK for a background pianist to use sheet music?

Why not? You’re not giving a recital or performing in a concert hall. Go for it.

The only difficulty might be not being able to play requests.

It doesn’t bother me, but make sure that you are confident enough in the music to not completely rely on the sheet music. I have to say, however, that improvisation works really well for background piano, since you are not the main focus in the room or setting.

Absolutely. Many background pianists use music. Others have been playing so long they know everything. But don’t worry about it. That should not stop you from getting a gig.

Sounds fine!

Seems fine to me. It’s better if you have it in front of you so in case you forget some notes, you’re not fumbling around on the keys.

Sure.

Totally acceptable in my book. I’ve seen pros at Carnegie Hall bring sheet music on rare occasion – although that’s usually for something madly complicated or ensemble pieces that are outside their core repertoire.
For a casual entertainer a big book makes sense. You could get a request for something you haven’t played in a Loooong time and if having the sheets with you makes it possible then, perfecto!