My state has been indicating, when queried, that college instructors and school staff that interact with students regularly are to be categorized as teachers. But yes, it’s a state by state thing,and within each school there may also be interpretation.
The vaccine is free, but if you get it at a doc office they can charge for the visit. Best to get through a pharmacy visit to your school or pharmacy direct,when possible (i think direct from pharmacies will not be possible during the first six months of triage..they won’t want to deal with requiring proof that you are an EMT or teacher, etc.).
Yes, states and municipalities, etc. can require inoculation. These always get challenged at some level, but never to the extent that they are voided. It’s probably tougher for an employer to do on their own, but I don’t think it’s blatantly illegal to try.