I would get another piece of roofing, and put that an temporarily, so that the run stays snow free. The way the coop is set up, the snow is going to dump onto the run. (At least i think thatis what it looks like) i wouldn't worry abut closing p the pop door at night, unless it is really super cold (negative) or if with the pop door open there is a breeze over the perch.
I agree about covering the top of that Run to keep the snow out of there. If you could add space I would do that too....how many hens is this for? Also. However, I'm a big believer in keeping any pop doors closed at night for predator protection. If there's a will and a way, they'll find it. And chicken wire (which it appears you have on there) A raccoon can force his way through or pull chickens through piece by piece if he has too. Sorry. Weasels of all sizes will find their way through that wire as well.