Turret Mode

When in Turret mode, Bunker cannot draw cards, but he can use two powers. Can he use his Initialize power to draw a card, circumventing this?

No. When a player is stated they cannot draw cards, they cannot draw cards. It has nothing to do with simply negating the draw phase; it simply does not allow card draws. Any action that would allow a card to be drawn, including but not limited to the draw phase, is cancelled.

Correct, unless a card states that it still allows players to draw a card when they wouldn’t be able to otherwise.

cough Ammo Drop cough

Just wait 'til y’all see Bunker versus the Matriarch …

Sounds excessive. Feathers absolutely everywhere.