1. If the ambient temperature is cold, and cold enough to stress your body, that could be enough to put your body under stress and make your nose and feet icey. Then when you hop into bed, you finally relax and start generating a lot of heat. Or it could be that you’re concentrating your fluids too much toward that part of the day. You could try eating less and drinking more around this time and see what happens.
2. No clue.