There is a port of the doom source code called ZDOOM that I really like.

Quakeworld runs nicely, even on slow machines. I recommend the port of the Quakeworld sources called QUAKEFORGE. If the machines don't have a 3D card, it can still run in software-rendering mode reasonably well.

The Quakeforge server software takes some configuring before it does decent deathmatches (you need to set up a config file for it), but it's nice because you can run the client and the server on the same machine. Then you get all the benefits of Quakeworld and don't have to sacrifice a computer for the server.

If you need help configuring these things, let me know and I can send you copies of my config files and batch files.