I've seen in the official QA that gog and steam should have crossplay, and I managed to play with my steam-using roommate through LAN without issue, but when I tried to direct connect to a different person (not on LAN) also using steam, it gave me an error using both IP and direct connect.

I bought the game on GOG specifically because I read that I could have crossplay between the two versions, but if I can't play with my friends, then that's seems really unfortunate.

Crossplay should work, so it could be a problem with your computer or direct connect in general. In case of the latter you'll have to wait for a patch, but if it's your computer these threads might help: https://www.gog.com/forum/divinity_original_sin_2/still_no_direct_connect and https://www.gog.com/forum/divinity_original_sin_2/cant_connect_to_friends_game

We did a GOG/Steam crossplay game last night with no issues, without using a virtual LAN of any kind. We did the following:
1. Player 1 (Steam owner) started the multiplayer game
2. Player 1 (Steam owner) went to whatismyip.com and found his external IP, and read it off to the rest of the players
3. Player 2 and Player 3 (both GOG) went into Story > Multiplayer > Direct Connect, and simply pasted in the IP

That was it. No problems at all.

Cross-play works fine for my friends and I.

Player 1 [Host]: This is me. [GOG]
Player 2: [Steam]
Player 3: [Steam]

I simply go to load campaign (once we've started one), open the main-menu[ESC] and click 'Connectivity Menu', make sure "Direct Connections" is ticked, click the blue button to right of it to copy the Server ID to your clipboard and paste it for them to use.

I never give them my IP address. Have always used Server ID without issues. The only time I've had a connectivity issue was when I forgot to disable my VPN before they tried to connect. So make sure any VPNs are disabled BEFORE launching the game at all.

The only thing that could be causing issues for you is if the host of the game doesn't have his ports forwarded correctly or there is some other issue with his network settings.

Please clarify what you mean.

Do you mean you're simply connected to the same router? If so then there definitely should be no issues as the game has LAN support by default.

The other way I interpret what you mean is that you're sharing a connection from one machine to another.

As in Router -> Computer -> Computer

If this is the case I cannot see any issues as long as each machine is receiving its own unique local IP whether via DHCP or a static one you've assigned.

I meant shared internet, not shared internet connection. So every PC has its own LAN cable what leads into the router.

My question is if I have a cross-platform game that works on an Xbox which I bought on pc can I install that game from my pc to that Xbox. Or do I have to buy it on said Xbox?