Agree, agree, agree! She's molting all right. We had one last year molt in December. She looked like a Naked Neck. She looked terrible! But, she's fine. Your girl will be fine. And, she won't lay while she's molting and for a little while afterwards. They use too much energy to regrow feathers to also lay eggs. Once her feathers finish out and she seems to have them all back and looks normal again, watch her comb. It's probably pinkish right now. When it gets nice and red again she'll start laying within a few days.
We don't heat our coop. And, the heat/cold doesn't have anything to do with her feathers. They can do pretty well in the cold. When it gets super cold, below freezing, we close the little door to the hen house at night after the girls have gone in and then we open it up in the morning. It keeps the draft out. If you do close them up, just make sure there's proper ventilation. We have little window flaps that allow air to come in and out, but not where it would cause a huge draft.