Will they like it if their nesting boxes are under the roosts on the floor?
Floor height nests are fine but I wouldn't have them right under the roost unless you're putting poop boards overhead to protect from... splatter.
Even with my coop closed up pretty tight It has gotten cold enough for some of my chickens to have gotten their combs frost bitten. My birds love to roost up in the rafters. It’s pretty airy up there.
Frostbite is moisture + cold. By "closing it up tight" you're causing moisture to build up in the coop, which can worsen frostbite. Hence why it's important in most climates to have ample ventilation available year round.
Obviously if you're talking -25F or more then yes the cold itself becomes a concern.