Today, I am happy to announce that my iPod has been reborn!
After work yesterday, I spent some more time trying almost everything suggested on Apple's support website (I was not going to re-install iTunes). I got a little farther than I did yesterday morning, but still couldn't get it to work. So I searched the internets for a bit, looking for non-Apple suggestions. I finally stumbled upon Relevant History: Meeting the sad iPod icon, which included the suggestion of dropping the iPod on your desk from about three inches. Sounds crazy, but I figure it's worth trying anyway. I mean, really, with the clicking noises, I figure the hard drive is pretty much shot, so dropping it isn't going to make it worse.
So I dropped my iPod on the floor from about three inches. And the iPod started working again.
Crazy. It took about 2 hours to re-populate my iPod with songs, as the things I had tried before had deleted all the data from the iPod. I'm listening to the iPod now and it's working fine. I don't *really* expect it to last, but every day I have with my iPod now is a gift. Besides, if it dies again, I'll just try dropping it again.