Warm air rises. I would not want open tops on sleeping boxes.In a cold climate, you will get frostbite on the males' combs. We've been in the low 20s here, and I've had to close the poop doors to keep my boys' heads inside.I have a huge cockerel who likes to sleep in the poop door way, and keep watch. With the door shut, he's been snuggling with his girls. His dad, Monty, always snuggles, so no problem with his comb.
No problem with moisture in my sleeping boxes. I pick, and fluff the deep straw every day. It stays clean, and dry with Sweet PDZ under it.
I was trying to remember your sleeping boxes. Are they deck storage boxes that have been modified? With several birds in one of those, do you have any fresh air inlets? Is it totally closed?
Our temps have been in the low teens at night here. I have a south facing door that is covered with wire and I have several pop doors for the rooster pens running along the eastern wall that I have been leaving open nightly about one inch. So far, so good, this year.
I plan to refigure my barn space and cull down to 4 large fowl males and keep each with a small flock of hens in a larger space in the barns and more space in their yards. No more roosters in single pens living alone waiting for their turn to be with the girls. This should reduce my work load quite a bit after I get it all finished. In the winters, I will provide sleeping boxes.