Does workshop provide any unique bonuses

The only adjacency bonuses are exactly like they were in the previous game: directly vertical and directly horizontal. For this reason, you'll want to build any workshops in the middle column of the base, rather than on one of the edges.

Once you've staffed a workshop, you gain access to staff those adjacent rooms with 2 Gremlins that are themselves engineers. This will net you once bonus engineer per workshop.

Obviously, this means that any adjacent facility that can't use engineers (Labs, for instance) will never benefit from workshops.

No, doors provide no cover. Once opened they remain opened even if closed.

I found this when playing a UFO mission and two soldiers either side of the UFO doorway. I opened the door to 'spawn' the enemies inside, which they did. Then I closed the door, hoping to limit their ability to hit me and force movement.

The next turn a Viper pulled one of my troops directly through the closed door with it's tongue attack, and ADVENT troops shot through the door at my other squad mates.