1) Am I correct in thinking no one makes .31 caliber gas checks?

2) Can I use .30 caliber gas checks on .311-.312 bullets? (And will they do what they're intended to do on such bullets?)

Yes, "30 cal" checks will work with .308"-.31+" bullets.

you can probably use the 30 cal GC's on up to .315 dia bullets.

32 cal GC's would be the next step for 8mm's of sorts.