Is it possible, that a friend of mine can play with me in LAN, if we both have the main game at gog and only one of us has a DLC. Can for example play my friend with me the Ashes of Malmouth DLC if only i own it? Or do we both need to buy the DLC?

If you have AoM installed then yes your friend needs to buy it as well to play with you, but if you don't have AoM installed then you can play together.

Both need to have the expansion or you need to disable your copy and just play the base game with them. And no character that's been in the expansion can be used for base game multiplay.

https://forums.crateentertainment.com/t/play-with-friends-although-not-everybody-has-the-expansion-s/86357

https://forums.crateentertainment.com/t/tool-gd-switcher-v1-2/51259