May have mold behind walls/ceilings not sure (leaking roof (crawl space) for LONG time-now fixed), musty bad smell in single family, dusty, 2 cats (male/female) in basement, sleep on 1st floor. Can afford to remodel this year. What air purifier you recommended?
A: That's a different situation than allergies alone, although the mold/mildew will affect allergies of course. The wall and ceiling problem is beyond what an air purifier can do, except for keeping it from spreading more and protect your health in the meantime.
There are three things to consider (and this is what I would do if it were me):
1) HEPA removes airborne mold spores to keep them from spreading (but doesn't kill the mold) 2) I would more often suggest the use of an inexpensive ozone air purifier which will do a good job of killing the mold. It just won't penetrate the wallboard completely and kill the mold inside. But it will reduce the mold enough to reduce some of the health risks. (keep it at a low level as too much ozone can irritate lungs). 3) Reduce the moisture (dehumidifier or A/C) as much as possible as mold needs dampness or humidity to grow.