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 about 35 degrees farenheit, maybe 40 at most. You don't want your chickens to molt and loose their nice feathery insulation because it felt like spring! And you don't want them too hot at night, either.
How cold does it get in the winter at your place?
As long as they have a dry place with not wind, chickens really can do fine down well below freezing.
Just for comparison--look at all the little tiny wild birds that live outside all winter. And think about people using down blankets and down jackets. Feathers really can be great insulation!