How do the bonus perks from the Advanced Warfare Center work

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.

• The attack is only triggered by enemy movement. Moving on your turn will not trigger Bladestorm (unless your move causes an enemy to move, such as a burrowing Chryssalid).
• No, only the movement will trigger Bladestorm. I've had an Andremedon move past my soldier (triggering Bladestorm), then in a separate action, make a melee attack. The attack was not responded to by my Ranger.
• They happen immediately, usually before the enemy gets to take their action. This is very useful against, e.g. Stun Lancers, which will move into attack range but die before they can actually make an attack. Some enemies, such as Faceless, have a melee range larger than the Ranger's, and don't actually have to move close enough to trigger Bladestorm to attack. (It seems there may have been a recent patch affecting this; it's possible for Bladestorm not to trigger until after the enemy gets to attack.)