Two trips to Home Depot later, with scarred hands and little metal screen shards on the porch, we managed to make screens for our windows. Does that mean we're now be-screened? Or just screening?
Regardless, we now have a way to admit the air without the creepie crawlies. We now have a way to keep the cat in and hopefully circulate some of the air out.
A word of advice: never ever rent a place without screens.