You have two temperatures to worry about.... minimum operating temperature and minimum temperature before damage.
Most electronics can operate down to 0C with no problems, some as low as -40C... but that's getting a bit rare now... Things like LCD displays will stop working around 0C, but will not start to suffer permanant damage until -15C ( lots af variation between units here)... Then there is the question of what happens to your lens as the parts contract in the cold.... they can deform if they get too cold and suffer permanant damage.
When your camera goes outside it cools off.... the less insulation around it, the faster it cools. Lens sleves will slow down the process, sticking it into an insulated box REALLY slows down the process.
I use a pelican case when shooting in the cold. When it is time for the shot the camera comes out... CLICK... and back in the case. Before I come inside I take the gear out of the case and put it into individual plastic bags, go in, and let things stabilize for an hour or so before opening the bags up.
If you are in a ski hut, there is a lot of moisture tracked in with snow, and more from cooking. Try to find a bedroom or something far away from the moist rooms to let your gear finish warming up and drying out.
and gloves.... get a light pair of gloves for shooting and a really warm pair of mittens to use afterwards. Holding a camera in the cold is very hard on the fingers.
Hope this helps....
NOTE: Picture shot near Resolute, Cornwallis Island, in the Canadian Artic Islands at -60C