Is there any way to play this game in co-op with steam user? I bought it here, friends bought it on steam. Seems to be no way to cross-connect.
If not, is there a way to get a steam key for it?
If not that, any way to refund the game?

Unfortunately, there is no crossplay. Dying Light Multiplayer on GOG is only available between GOG users (other games offer crossplay, the choice is up to the developers).

And no, there is no way to get a Steam key on GOG.

If you want, you can try to ask for a refund here.