After I cooled off a bit I decided to try one last thing. I yanked the ethernet card, a 3COM 3C905, out of my old box and installed it in the new computer. Win2k saw it no problem.
From there I proceeded to patch immediately. After that, I found the drivers I needed on the NVidia site (it's an NVidia motherboard of some sort - or maybe just components - I'm not totally sure). Anyway, I got video and sound going and even got the integrated ethernet working! I disabled it, though, in favor of keeping the 3COM card going. I still don't think the integrated port will work with Linux.
So, short of hardware failure, everything looks good now.