Yes.
(As long as the wind isn't blowing directly in the door all day--if it does that, put up a wind break to block some of the wind before it reaches the open door.)
I wouldn't, unless it's controlled by a thermostat. Seriously, I think you should NOT heat any part of the chicken coop above...
I suggest you put a thermometer in the coop, and base your decisions on the inside temperature reading. (Because inside temperatures and outside temperatures do not always go up and down together.)
I also suggest that if you add heat, only warm it to "above freezing," not actually warm. (So 32...
What breeds of chickens?
How cold?
Most breeds of chickens do quite well in unheated coops.
If you do not use a heater at all, they will gradually get used to the cold, grow an appropriate set of feathers, and will usually be fine. If you use a heater for a while and then stop, they will not...