With that said, can I possibly only allow traffic to and fro from the proxy through my firewall?

Yes. That is what I suggested. If you configure the firewall to only allow traffic to/from the specific IP and port combination of your proxy, other traffic will be blocked.

I should be able to (in theory) inspect traffic too, although I don’t know how far that will take me.

You can do content filtering via a proxy like that, yes. A similar sort of configuration is used on school computers to do things like block adult content, with varying degrees of success. Some ad-blocking techniques work on similar principles.