Good idea to replace that door. Not everyone has the skills to do that. The farmers around here use your shade screen technique on their cow barns. Cheaper than using wood or metal all the way up, and gives great ventilation in the warm weather.
You could raise one end of the thing a few inches. This would tip the roof and encourage the snow to slide off. Otherwise, you could create second, independent, roof over the first. This second roof could be tipped by you and a friend after a snow to get some of the snow off, then use a snow...
FIFTY! years ago? You must be really old. I'm really old and I remember the neighbor kids had one of those cars.
If mud is a problem you could put in French drains or something similar. Sow with a mix of things that chickens will eat and that chickens won't eat. They don't seem to like bluestem.
Rancid, I like to put down straw over the mud. Sometimes, more straw.
I am in Michigan's thumb and it doesn't get that cold. If the insulation is overkill, you can always add more ventilation, maybe a fan.
On your nest boxes, I would have the door open on the opposite side so that it drops down, not up. This way, you don't have to hold up the lid when you get the eggs.