Thank you, everyone, I am already feeling more confident just with
your guidance and input.

Here are a few variables:
"Tax-return scared unless it's a very old house": Uhm...does 120
years count? Yeah, seriously, it's an old "farmhouse Victorian."
Because it's old, I'd like to stay away from more modern solutions
such as tub surrounds.

I have shopped and asked around quite a lot; for some reason, it's
surprisingly difficult to find a "tile guy" around here. Everyone
seems to sorta know someone who sorta knows someone but can never get
me names/numbers. I've found just two guys in the area so far.

However, I did NOT think about asking people how their bathroom
remodels went, rather than asking "hey, do you happen to know a tile
guy." Sounds like a good approach and one I'll start using.

There is absolutely no doubt in my mind that there is already a mold
problem. On the third wall (the one I see as the problem wall,
although from what you say, it sounds as if the other two might get
involved) there is obvious black spotting and plaster crumbling at the
bottom by the corner of the tub (not real plaster, plaster from the
drywall). I just hold my breath when I'm in the bathroom (ha ha). I
do totally understand and appreciate the issues with mold, one of the
reasons I don't want to keep delaying this.

I definitely will be replacing the shower fixtures when the job is
done (already all set with a reliable plumber, thank God. I
practically have to keep him on retainer with this house.) The faucet
leaks now and although I got some info from Peerless about repairing
it, the parts are no longer available and it looks like a cheezy
faucet to me anyway, so I'm opting for replacement. I've already
replaced a couple faucets elsewhere in the house and priced shower
setups (mixer et al) extensively, so I know exactly how scared to be
about that (picture very sad look on my face, as I have learned I seem
to have expensive taste in faucets. My plumber is also rather
specific about not letting me buy crap, and I figure he oughta know.)

It's very nice ceramic tile that appears sound. I would really like
to reuse it if I can, both because of cost and also because there is
some other tile work in the bathroom that matches it and is without
problems. There are a few spare tiles squirreled away in one of the
utility cupboards.

I'm 100% certain all three walls are drywall/sheetrock (unclear about
the difference, if any), based on previous experience with other areas
of them. The other two really look sound, but I guess we'll see what
we see. Is there a way to tell if there is some kind of backer
board? One small (1 x 3, maybe) tile has fallen off the far wall; can
I look at that exposed surface and figure it out?

I would not be the least surprised to find that corners have been cut;
I've found that many many (MANY MANY) times elsewhere in the house.
It's always kind of an interesting history to take on any project,
because I can quickly see whether the last person to work on it was
one of the owners who really took pains with it, or one of the owners
who did "whatever" to get a cosmetic, but not correct, outcome.

I hadn't considered doing the tear-off myself -- GOOD idea and I'm not
afraid of doing it at all. Even have a Dremel that will supposedly
grind out grout and would probably be useful cleaning up tiles (I'm
assuming the old adhesive must be ground off --?)

Alas, it is the only bathroom. I like to camp, though, so I know how
to rough it if I have to

Thanks again! Will report back as the project advances.