Is going to school and re-classing to a different MOS considered a TDY or a PCS?

It really depends on the course. Courses that don't produce an MOS and return you to your unit are usually TDY. But courses that produce an MOS and an follow on assignment are usually a PCS. An example of this for me would be my Captain Career Course. I would PCS to Fort Benning for 6 months and belong to the School. When I finish the course I will then report to a new unit. The reason why this happens is that when you are TDY you still belong to your original unit. There really isn't a reason to do that if you are not going back there. Plus you could imagine the nightmare if you needed some admin stuff done.

You’ll be in barracks more than likely. Long term reclass schools they’ll have barracks set up. You’ll receive a meal card and lodging.

180 days normally I thought is a PCS. Billets should be provided