Recommendation -> don't go from cold to hot and then back to cold. The humidity in the air and in your camera will condense into water and possibly mess it up. A lot of electronics have some sort of sensor that will automatically tell you that there is water/condensation in the camera and shut it down.
Basically, once you bring the cam into the lodge, leave it there. It just makes things easier. Eat on the lift/outside, or leave the cam in the car while you eat? Either way, keeping it cold may take more work but it solves some problems, at least for me.
As for lens cleaning, rubbing it with basically anything except some sort of lens product will probably streak it. Maybe try blowing on it? It will get foggy but you'll blow the beads off and the fog should go away...