The first and most recent coops I built have a slanted roof. It really wasn't that difficult, and liked that it put the rain/snow run off in a predictable place - away from the run. When I put in the first slanted roof, I didn't even cut angles in the roof supports. I just built a tall wall and a shorter wall and laid the 2x4s across them and attached them with a couple of deck screws.I much prefer flat roofs! slanted roof would be weird since one side is 5 ft. and the overhangs would be near the ground. anyone who has a flat roof and gets snow let me know.
Here's a pic of that first coop that kinda shows the roof. Kinda. But you could do the same on a run.