Yes. GOG has achievements, GOG Galaxy must be running. I have Witcher 3 on GOG, and there are achievements.
Yes. It's easier to play multiplayer with steam. But, you can play cross steam-gog multiplayer via direct connect with the gog version. The lobby in the gog version is not working, yet.
Not all games has achievements in GOG Galaxy, it depends whether developer decided to implement them. Divinity: OS (EE as well) doesn't have Galaxy achievements.
