I have a few thoughts...
1. My queen banks are always in a queenless hive. I don’t think it would work otherwise.
2. I am sure overwintering is possible given enough bees, but I would be concerned here in WI with the bees being able to get to the stores and still be able to feed the queens as winter progressed. I would assume you would run into the same issue up there? What is the plan for emergency feeding if needed?
3. As far as when, anytime, but if you are concerned About injuries etc I’d wait until as late as possible to install them, and then still be able to go in and check on them one or two times after that (more temp and weather dependent in your area rather than calendar date I’m afraid)
4. I have seen screen used in the past, so assuming that should be fine?
As iddee suggested, check with Perry- he is probably the best one on this topic due to location and expertise with queen inspecting...
Good luck!