We finally managed to get the home warranty company and the contractor to talk. Apparently, though the furnace was installed with the cooling coil on top of it - which both contractor companies say reduces the efficiency and the lifetime of the coil - the manufacturer does not require the coil to be beneath it, so the home warranty company will not cover the replacement of the furnace or the work to correct the issue. So, the contractor is only going to replace the A/C; we figure when the furnace dies, the home warranty company will pay for it then.
Thus, we have an appointment for the work and we should be back living in our house next weekend! They did specify that they needed access to the crawlspace, and that's quite a mess. The access is in my bedroom closet, and therefore underneath a ton of boxes and behind a ton more boxes and mess, so we're spending the weekend and the evenings of next week trying to dig out a path to it. On the bright side, I've been putting off doing this - cleaning my room - for years now, so it should be awesome when we're finally done. The goal is to have a nice desk in there where I can write and do crafts and such, and if I'm really ambitious, I'll replace my dresser as well.
We are so itching to go back home. It's been great staying with our friend, but it's long past time to get out.