I don't worry about keeping them warm. Like the wild birds that overwinter there they have feathers and down to keep themselves warm. To get rid of the moisture I'd open it up where you have it marked and cover that with hardware cloth. I think it has an overhang to keep most of the rain out. On the side under your covered run I'd open it up and put hardware cloth there up high. That should let in enough air to handle moisture with any breezes going over their heads when in the roosts.
