Hmm...very interesting. I reinstalled (actually, slightly upgraded) the kernel to 2.6.32-25-generic. When I do a modprobe snd-intel8x0, it now does not return an error, but /lib/modules/2.6.32-25-generic does not contain that specific driver. It has:
acore core drivers i2c isa oss pcmcia soc soundcore.ko sound_firmware.ko synth usb
A locate finds the snd-intel8x0 in older linux directories under linux-2.6.whatever/sound/pci.
Sound still doesn't work in KDE after doing the modprobe, and trying to test ALSA in system settings crashes the app.
[ 5856.260117] Intel ICH 0000:00:02.7: PCI INT C -> GSI 18 (level, low) -> IRQ 18
[ 5856.600024] intel8x0_measure_ac97_clock: measured 50761 usecs (2442 samples)
[ 5856.600031] intel8x0: clocking to 48000
So, in short, it looks like the module loaded and found the sound card, but the desktop can't use it.
Thanks for the tips so far, very helpful!