I tried to play co-op with a friend but nothing seemed to work. It doesn't matter wich privacy settings, game mode or host we choose, all attempts fail instantly with "Matchmaking failed" for the player trying to join. We couldn't try the invite code, because we don't know how to use it.

Was your friend on steam by any chance? I had the same issues when connecting to any steam player.

Also for the invite code:
When you created a lobby go the the invite friend option at the bottom there should be a button that says create code.
The other player goes to quick match and insert the friend code there to connect to the lobby.