do rear facing car seats have to have a base (by law) in ohio?

Answer 1

No. They just have to be strapped in securely.

I live in Ohio, and when my son was in a rear-facing seat, the seat part had slots for the seat belt, though we did mostly snap his seat into the base. It's handy, though, so you don't have to keep adjusting the seat belt when you put the base in another car.

Answer 2

I can't say positively, but I am pretty sure it does.