If a professor used a particular graduate TA for their class, would it be possible for that professor to use the same TA for a different class the following semester?
It’s definitely possible. I’d say it’s even common in many instances.
Probably depends on the school, but I don’t see why not, unless that grad student doesn’t want to or can’t TA again. I’ve TA-ed for the same professor as an undergraduate multiple times but it was for the same course.
Yes, this could happen. In my department, the grad advisor gets involved in the allocation of TAs to professors and we don’t always get to choose anyone we want. Sometimes a grad student has classes at the time that some professors are teaching, so not all grad students are available to TA every class.