part of the weird is the shared AC system should be pumping fresh air. I mean it sounds like cold air is coming in a tube to all rooms from a central place.

against cold air the solution is to put up some shirts. why aren't you doing that?

why do you believe the AC system is spreading covid? is covid an active threat right now in your area?

I don't have solution to the cats, maybe cat doors. but what will you do when winter comes and it will be cold outside?