Thank you all for your positive comments to my slightly hysterical post yesterday! Seriously, people were coming out of the woodwork (I'm looking at you, zenchick). Anyway, here's what happened:
When we found out on Tuesday that Installation Guy had quit (which reminds me, we need to change the locks, he still has a key), Sales Guy said he'd come out yesterday to assess what needed to be done. Once he'd seen how much was left to do, he'd be able to schedule a time to come out and finish. This was totally fine with us -- we know the situation IG's actions have left poor Sales Guy in, and it's not a pleasant thing.
So he comes out in the afternoon, looks around, and says, "It's pretty awful in here. Let's see if we can at least get the air conditioner turned on."
(If you could pronounce exclamation points, I would have at that point.)
So he banged a few things, fiddled around in the basement for a while, and lo! The air conditioner was turned on! The job's not done, mind you -- ducts still need to be run to the kitchen and bathroom, and one or two vents in the living room aren't hooked up, and one of the returns isn't installed yet, but most of the vents on the first floor are in fact blowing cold air. He'll come back as soon as he can to finish up the ductwork, but in the meantime we have blessed, blessed air.
It's not really cooling the house down, since there's no insulation right now and no walls to separate the rooms, but it's performed a huge service by getting rid of the swamplike humidity on the first floor, which has already made our lives SO MUCH BETTER. The cat, last I saw her, was sprawled happily on the living room floor, enjoying the change. I may join her.