I have a Hoop coop, made from two cattle panels and tarps (inside the run). they roost on a "saw horse" inside and the hen lays in a box on the ground. The whole front of the shelter is open. We don't have real cold temperatures here, but I have read several that say theirs sleep outside even in the cold. But they do have to have a roost and prefer a 4x4. 2x4 is too small.
I built a 4x4 coop with approx 10 x 8ft run. they seem to like it...but they love it when i let them out to graze. I am afraid the 4x4 pen may be too small, thinking about an add on. I read somewhere they like 5 square feet...if that is the case them i am alright with 16 square feet.
I am sure some one here has a better answer for you, as these are my first go at the turkeys