Shower Valve Handle set screw jammed

Hope you all can give me some insight. Tennant called and said the handle from her shower keeps falling off (I just purchased the property this summer). Taking a quick look the other day while I was out there I see there is a set screw on the inside of the handle that appears to tighten against a valve stem extension... however I'm just guessing. I know there is a set screw on the outside of the handle and one inside. Here are my questions:

1) Does anyone know what kind of valve this is? It's old and heavy. I don't see any manufacturer.
2) Do you think that set screw on the outside of the handle separates the handle in two pieces so you can reach the inner set screw?
3) If the design is how I'm thinking, I suspect that valve stem extension is probably corroded into the handle. I can probably get it out, but how would I affix it to the existing valve stem mounted to the fixture?
4) Lastly, do I have this all wrong?

I appreciate any insight. Thank you!