How do I replace switch/outlet to GFCI/switch combination?
My suspicion: an open ground

This is based on two factors:


Your three-lamp tester is unable to trip the GFCI, but the GFCI tests OK.  This is because the internal TEST button doesn't rely on the ground (it goes load-hot to line-neutral or vice versa), but the three-lamp tester's TEST button is connected hot-to-ground, and thus requires a solid ground connection to work.
Pushing the TEST button on your tester yields a "hot/neutral reverse" indication despite it testing correctly initially -- the "correct" indication is due to capacitive coupling between neutral and ground providing a current path "home" for the current through the right-hand lamp.


I'd check to make sure the green wire is securely connected to the box in your case; if it is, then the problem's upstream somewhere as it appears this receptacle was wired using conduit (note the locknut and fitting at the bottom of the box).  If you can't find it still -- call in an electrician, as they should have better tools (such as an earth fault analyzer/loop impedance tester) that can be applied to the problem.