Tile Shower Repair advice needed

[brutal]

I need to do a little repair in the master bath shower.

4'x3' shower with an acrylic(?) base. 6x6 Ceramic tile walls, sliding glass door. Old brass hardware.

Some of the caulk (more on that later) started to discolor/mold. We had two tiles low and outside the door start to pop out. Appears to be a poorly done previous repair.

I removed the door and rails, etc. and have a couple 3-4 edge tiles safely removed and one full tile from the bottom corner I haven't yet removed (so far I think) that needs to be redone.

My plan is to cut in as far as the backer board (only drywall?) is viable and patch some blue/green board in and fiber tape to rebuild the backing. Then reinstall the couple tiles I pulled off.

I plan to grout saw the rest and regrout it. Not looking forward to that and may have to buy another power tool to do it.

I may clean up the doors and reinstall (most likely at this point), I may buy another. Stuff is old, bright brass. Modern brushed nickel frameless hinged door set (46") looks to be about $600-700

First question. The corners and bottom have "caulk" that is rock hard. It's not grout. WTF is this stuff?

Second question. What is the recommended product to use in the corners and bottom where the tile meets the base? Personally, I'm not a fan of Silicone caulk. Primarily because nothing sticks to cured silicone, even fresh silicone. Seems anything that's done with silicone, should it ever need to be redone, is a mess and never seals properly again.

Third question. While I'm not really ready to drop $$$ into it right now because I can't really afford it, and not sure I'm up for either the tile work in skill or time, although I have done it in the past, what might an approximate cost be to just redo the whole thing assuming I do the demo and put up new backer and do most of the prep? It's about 80 sq ft. Looks like about $10/sq for the tiles alone?

Open to ideas and suggestions, but primarily looking for help on questions 1 and 2.