KAK - there's a thread in Cat General about tons of dust and stuff - some people have filtered their vents (if they have forced air heat) to filter out dust. That might help - in addition to a furnace and duct cleaning.

I used to use a layer or two of cheesecloth. The air still got through, but the dust didn't.

Might help Sky?

If you can rent a carpet cleaner cheaply (or borrow one), maybe give the rug a once-over. I know you are getting rid of it, but this might help in the meantime.

Good news on Starr! Hope Sky feels better soon, and the Storm needs to change his name!