The first problem you have is a store bought coop that is way too small for the number of chickens you have. However, you'll have to deal with what you have. Chickens do better in the winter if there is 4 square feet per bird. Overcrowding sometimes causes the birds to fight and pick feathers. So for your 6 birds your coop alone should be 4 ft by 6 ft.
If I were you I'd put clear plastic up on 3 sides of your run. That way the girls have a place to come out to that is free from wind. If your run isn't covered in some way I'd try to figure out a way to get it covered. Many chickens do not like to be out in snow. Mine refuse to walk on it and I live in Montana where we have snow on the ground 5-6 months a year.
The main problem you will have is getting the moisture out of your coop. A dry chicken is a warm chicken. You really don't have might height with that coop. I know because I started with one just like it years ago. If you use clear plastic on the run you should be able to open up the coop up high on the side that faces the run. That should help keep moisture from pooping and breathing to vent out of the coop without having a breeze blowing on the birds.
There are various way to handle the water freezing in winter. I use a container, mine is a tote, horizontal nipples, and a stock tank deicer that is rated for use in plastic. You could plan on replacing the water twice a day. They do sell some heated waterers. What you choose to do depends on if you have electricity to the coop and how often you can go outside to tend to the birds. My water is kept in the run to reduce the amount of moisture in the coop.
I worried the first winter with the chickens. It all turned out okay. Good luck.