Zandronum

I have quite a collection of old laptops going back 20 years and am wondering what the easiest way to get them all playing Doom on the LAN together would be.

Operating system
I'm thinking because of their age (one is Pentium 3 for example) this would have to be a lightweight linux. Is there a distro that is lightweight and works well with Zandronum?

Doom variant
Is Zandronum the best choice here? I read that it includes modern updates which might make the old machines choke.

AFAIK any linux distro that isn't broken should work with zan.

I don't know if zan is the best choice. You can try whatever you want to play with the port on the least performant of the laptops and see if they can handle it I guess. There is also the choice between using software renderer and OpenGL renderer, which is a big deal in terms of performance.
A possibility would be to use an older version of zan or to go all the way back to skulltag. You could also look at other multiplayer source ports but -i don't know anything about them so I can't help you there.