Hot air rises, cold air sinks, so the screen bottoms in this respect should not be an issue for over wintering. The condensation moisture inside the hive in winter conditions is not as big of issue with screened bottoms as the water flows right out the bottom screen.
Breezes and winds are another matter, they could cause cold air to move up into the hive, I would imagine, in an open screen bottom. Puting a piece of plywood over the screen or leaving the sticky board in over winter might be an option.
20+ years, raise my own queens, feed when needed. I treat, but have not perfected varroa management yet.