My son, who tens to build very solidly, built our 11'x17' coop similar to yours. The roof slants from one 17' side to the other. He had some (sort of free) strong scrap metal to work with, about 2" x 2" square "pipes" which he used for framing. There is one pipe running across the short side of the middle of the roof area. Across that and the sides he put 2x6's about 2' apart. Over those he put metal roofing, so the 2x6's were placed so you always had a place to screw to. I don't see why you couldn't do the same thing with wood; you might want a doubled 2x6 for the single support across the middle of the short direction. By doing it this way, water runs off the short side well. Ours is several years old and there is no sign of sagging. The metal roofing was actually more heavyweight than we would have chosen -- but again, the metal was there, "sort of" free. If it didn't seem solid enough, it would be a simple matter to put one upright smack in the center of the coop to support the middle of the roof. Might even do this and just put one 2x6 across, instead of a doubled one.
I would cover the two end triangular areas entirely in hardware cloth for ventilation (1 sq ft per bird is the minimum.) If you don't get snow load, you will need the air space, I promise. Ours is open to the air on all 4 sides above the walls (like you, we also built the roof with some overhang, thank goodness.) Ours roost back in a corner to stop drafts on the rare night that a winter draft might actually disturb them; most of the year, they aren't drafts, they are welcome breezes. We have a framed hardware cloth people door, and part of one wall opposite the people door is all hardware cloth. In my climate, cold is rarely if ever a problem, but heat is a challenge. We run a box fan all summer, mounted up high, more or less where your largest end of your triangle is, blowing out. The chickens will use the coop for midday shade on hot sunny days, it is so comfortable in there.
