Some do, some don't. If you don't, you can expect stiff, frozen boots in the morning. While having them in the bag can be a little uncomfortable, those who do will have toasty boots. In their own bag and arrange them someplace in the sleeping bag where it's comfortable. Moist mittens and socks can be dried by placing them underneath your shirt next to your warm skin.
Gaiters commonly freeze up.
Before retiring, knock the snow off and place them under your bag. Or put them in the same stuff bag as your boots and take them to bed. If you have problems with the zipper, rub on snow seal or candle wax. The first person awake in the morning should bravely get up and fire up the stove. He or she can get the others going by serving them hot drinks in their sleeping bags.