Do not think warm, think dry. Dry birds protected from wind are warm birds. Birds trapped in a small coop with no ventilation, in the attempt to keep them warm, instead are damp, and damp birds are cold birds. Think of being in a car, in the winter with the heater turned off. Condensation is almost immediate, and that is what you are trying to stop from happening.
You want to have space above your chickens, my vents on under the eaves of the roof, and I never close them. Measure how much space is above your chickens heads. Needs to be about 15 inches. Look at the set up of the roosts, how close are they two the walls? Especially the west and north wall, should have 15 inches cleareance too. This allows good air flow between them, and the surface, allowing moisture to naturally rise and leave the coop.
Deep dry bedding. Once a week or so, scatter scratch on top, and they will dig through it breaking up manure clumps, and turning the bedding, keeping it much drier in the coop.
Hope this helps, sending a picture of the coop, and the number of birds can help too. I do have a wind shelter out in my run. And I have bedding in my run too. If I know a storm is coming (Maybe this Thursday, yikes) I will sweep the bedding into mini haystacks, after the snow, I flip the hay on top of the snow. Encourages them to come outside.
As for the eggs, they will freeze. Sometimes cracking the shell, if it is a hairline crack, I use soon in baking, if it is a thick crack, I thaw and feed to my dog. If you can collect them earlier, you will have better luck, I work outside the home, and can't.