I found a very close hardware store, saving me from actually getting into rush hour traffic. The valve that didn't look like any of the descriptions, photos, or diagrams on the web? The guy behind the counter at the hardware store said, "Oh, that's not from a faucet, that's from a shut-off valve." After I confirmed that, he continued, "when those go bad they want you to replace the whole stem. You're not supposed to be able to fix those. But we can fix 'em. I'll have to put the washer on for you though, 'cause you'll never get it on there yourself."
Actually, I think I screwed up by forgetting a step ("try to find any remnants of the missing washer inside the valve (which is positioned where I can't actually look)") in my haste to get everything back together. But it appears to be working well enough for now -- the cutoff valve is acting a little funny but the faucet does its job, and sometime when I'm feeling more patient I can shut off the hot water in the basement again and have another shot at the cutoff valve. Until the faucet goes bad again, the cutoff valve shouldn't matter.
So now to wait for the water heater to come up to temperature, take my long-delayed shower, and then see whether I remain standing long enough to help the freshly doctorated (doctorified) housemate celebrate. I'm still running a big sleep deficit, so I may wind up falling over after my shower. We'll see.
(no subject)