I think I may have already made progress. In the BIOS Setup, I did a (IIRC it's called) "Force IDE device redetect": I pressed ALT+E then ALT+F, save settings & exit. When I rebooted my Ethernet card WAS recognized.
However, since my SATA cards have been recognized for months and I've had hard drive filesystem corruption with them, I'm inclined to try your great idea to reflash my BIOS and reinstall my chipset drivers.
Is there any way to test the problem slot? I don't want to put the SATA cards back in there (I'm sick of the filesystem corruption), and I don't know any way to test an Ethernet card or the slot it's in.