Spongy wall and paint is peeling- what kind of specialist do I call? How much is an estimate, then, the work??

I'd start by calling a plumber. He is gpoing to tear out that wall to fix your leak. He can then direct you to a handyman or bath remodeler who can install the correct wall material for a shower. The correct material will not get spongy but that is just a function of the leaking pipe insiode the wall.