We just got ..um.. "advised" not to run a LAN, but were told we could run a peer to peer

This is strange. They say it's OK to play games, but not OK to use the network to do it?

I don't understand. If it's because of bandwidth, tell 'em that these games use up incredibly tiny amounts of network bandwidth.

The very first version of Doom1 used broadcast packets which did tend to eat a little bandwidth, but they fixed that up almost immediately and it hasn't been in any version since then. Now any networked game will work fine on any network with minimal impact.

Anyone have any experience running Doom 1 or 2 over serial cable?

Yes, as you said, a long time ago. It only barely worked, and it was buggy as hell. You would get disconnected all the time, it was tough to connect, the cable had to be just right, etc.

The main limitation with a serial game is that you can only do two player. Not nearly as fun as four-player LAN action.