It seems to me that there may be less expensive solutions than
covering your entire floor with rigid, durable plastic.

You could run an air purifier in the apartment. I don't mean
something bogus like the Ionic Breeze that doesn't actually do
anything, I'm talking a real, HEPA air purifier that'll make a
difference in the quality of the air in your apartment.
Consumer Reports has reviewed air purifiers, so there is data
available on which ones do a better and worse job of cleaning
air.

As for the vacuum kicking up dust, you need to get a good
HEPA vacuum to solve that problem. I believe Consumer
Reports has reviewed those as well. There's also a ton of
information about HEPA air purifiers and vacuum cleaners on
the Web.