I want to play this one with my kids. Does it have LAN support? Does it require one copy per household or one copy per player? Thanks!

Should be able to play it on LAN if you don't have GOG Galaxy and it'll be copy per player.

Excellent. Thank you.

I believe this was 1 copy per player.

2 Players can use 1 copy. I use it to port my stash-chars (accomplished hero, lvl40) to Knossos (less walking to vault) with 1 copy on 2 (physical) computers.

The game has troubles binding to the correct network interface, so be sure, that you disable all interfaces (in the windows network settings) but the one you want to play on (LAN-Interface in your case).
In my case I had to disable all the VirtualBox Network-Interfaces to get it work - I dont use Galaxy.