Long story short: I bought a replacement water heater, a regular tank heater, and my dad and I installed it this morning. Here it is, all shiny and fresh. Note that it has a separate cutoff - which is useful, because of course I haven't replumbed the hot water yet.
I tested the hot water, and yeah, got kind of soaked. There's at least a fitting blown under the bathtub; the rest look like they might be OK, but it's hard to tell when all that water is spraying around.
But - as soon as I have that fixed, and call the gas company to get my gas turned on, I've got hot water!
In other news, I've made a lot of progress in mopping ceilings and walls. It's looking a lot better in here. Still smells like bleach, but that's a real improvement on what it smelled like two weeks ago. Ha.
And in other other news, I measured the bedroom window, just to get an idea of what it might cost to replace windows. It measures 36" x 54" - meaning it's a lot bigger window than it really looks in place - and wonder of wonders, that seems to be a quasi-standard size. I have a wooden option at Menard's for $250, and Lowe's has a vinyl insert for $150 or so. In total, there are six windows in the carriage house, and frankly, they're nothing special, so I don't feel bad about replacing them with something more modern. So I could do it with halfway decent wooden windows for $1500, or cheaper-looking vinyl for $900.