I wanted to show my son this game. (he's young so not expecting much.) But I wanted to play with him on a separate old laptop on the network or battle.net. But Battle.net requires CD key and the other options are confusing and just crash...

So this can go 2 ways:
Battle.net: Do I need more than 1 copy of the GoG version to use battle.net on the 2 computers?
Other Multiplayer options (not battle.net): What else do I have to do to get us to play? The other options for multiplayer seem out of date. Like calling on modems.

We maybe just want to try out 1 game together so kinda hard to commit to another copy when I used to own multiple in the past.

You can actually play with any Battle.net edition cd key so if you still have those good old cdkeys you can actually use those alongside your gog serial key :-)

Battlenet require two cdkeys.
IPX-IP allow interet or lan play same as Bnet up to 8 players. Other options are outdated even more than IPX (allow only direct lan connect pc to pc for two players).
Enable Legacy DirectPlay and IPX wrapper required, both normally should be pre installed with the game. Without them alternatives to bnet wont be available in the list.

Enchancment online in game. In launcher for lan play it can be any of classic/enchancment (in gog launcher) - but same.