I keep the food and water in the run, not the coop.  There really isn't enough room.  I use a heated dog water bowl, I just run an extension cord from the house and it stays liquid all winter.  The food is kept hung in a small shed we build from plywood. It's open, but it is sheltered from the weather.  
We have a minor problem with mice. I place a few mouse traps on the outside of the run.  But, the girls can eat outside and stay out of the weather, even gives them another place to hang out.
When it snows/rains, i scoop if the snow is deep, then I add some straw to floor to keep their feet out of the wet.