With much grumbling about Microsoft and how this would be so much easier in Linux, my compatriot J got the wireless network running. Our main technical difficulty: The used router I got back in '05 really wants to be connected directly to broadband, and we're running dialup through one of the PCs. So we had to go static IP instead of DHCP, just to get things up and running.
And, for some reason, the NIC on the motherboard on the PC with the dialup doesn't even show up in the BIOS, so it's talking to the router wirelessly. I found an old 10/1o0 card in "the pile", and may whack it in there.
But, by the end of the night, all the PCs we had on were able to see each other, share the internet connection, and share the printer. Victory! And the multi-player in HoMM III (the whole point of the exercise) looks like it'll run happily over TCP/IP.