It's probably nailed on, holes puttied and paint. Try slipping a putty knife in the seam and wiggle it back and forth.pull away from the cabinet with some force, if it moves even a little, it's nailed. Should be able to pull with hands

I'm not a plumber but I am in construction, if you ask me it's lost. You can shut the water off to the house and try to tap it out anyways. Looks like a fiberglass surround, if you open the sheetrock on the other side to access the valve you don't have to rip your shower out. Be prepared to replace it if you try to tap it out

What's it screwed in to? If its just a mounting screw you can tap it out with a tap out set and just get a bigger screw for it to install the new (screws at ace hardware, large variety) if it's a part of the valve construction, get a new valve and don't try to fuck with it