If I buy Dying Light: The Following Enhanced Edition here in GOG.com I will can activate the key in Steam?
No, you will get the game in your GOG catalogue. There do not appear to be any keys associated with the game in my account, so multiplayer I assume is handled by the Galaxy client.
I do not know if you can cross play with this game as I do not know anyone with this on Steam.
As far as I know, no. Since Steam multiplayer is handled by Steamworks and GOG multiplayer is handled by Galaxy. This is a consistent problem when devs rely on Steam tools for multiplayer/leaderboards/mods.
On one side Steam tools are nice since it saves devs some (or a lot) of work, on the other it makes Steam become an actual platform and not just a storefront and subdivides the PC platform.