A Zero would probably be way underpowered for the job. I've used a Pi 4 in the past and it worked ok, but choked occasionally.
My actual recommendation would be a small x86 box, something like a Lenovo Thinkcentre Tiny. You can get them used for about the price of a Pi, and they'll be much more reliable.
Exactly. Doesn't matter if they're wired or wifi, or where they are, as long as they're on the same network you're fine.