Thanks for the pictures, they are very helpful with your question. Nice construction of your shed, by the way. I like your use of pallets.
If your goat shed door is facing away from where your bad weather comes from, i.e. if your winter storms come from the North and your goat shed door opening faces South, then I wouldn't bother with a door.
From one of the pictures it looks like you have Nigerians? Small goats? If so maybe just put in a larger dog house like in the middle picture, something both could get into if they wanted. You might be able to pick up something inexpensive on Craigslist, like those Igloo type shelters. If you do that then I wouldn't worry too much about the slits in the walls... unless you find snow accumulating inside the shed. You can always put some hay inside the dog house.
I would not think it a good idea to stack bales of hay around the wall perimeter to block any wind as the goats WILL end up jumping up on the bales, laying on the bales, pooping and piddling on the bales and the poop and piddle will ruin the bales.
In my experience, unless you put up plywood on the outside to block the slits, anything you put up like a tarp or real heavy plastic sheeting will just get chewed up by the goats. They do HAVE TO put their mouths on everything... and then pull it.
Also... A big howdy and welcome to BYH from the Great Lone Star State, where the stars at night are big and bright. Glad you found us!