I happen to be allergic to planet earth

with or without pets

i also have a zoo. I got "allergy free" dogs, aka maltipoos.. oh and had hives for the first three months they were with me. That subsided. I do have several chickens and a parrot. Desert variety parrots are dusty like chickens, i have had them in the past. My current one is a tropical jungle breed, so no dust.
The chickens are in a room with hard flooring that can be vacuumed, swept, and mopped frequently. I also have an exhaust fan in their room that i run during day hours. I also have a honeywell hepa air purifier and filter next to their cage (and in all of my rooms, because i just feel healthier with them). I clean poop daily, using a kitty poop scoop. I created a skirt around the bottom of the coop so it contains everything when they scratch. I experimented quite a bit and settled on sweet pdz granules (granules is the key..) which is heavy enough to not go airborne, absorbs all ammonia, is light enough to not over weight your trays, and is safe for the chickens to eat as a grit.
The chickens get to roam for a little while.. 20-30 mins seems sufficient to them, after that, they just trek back into their coop/run. When i was sitting with them for extended periods, id use the chicken diapers, but with this shorter interval, i get no more than 3 poops. I just pick them up, then use pet spray and wipe up.
My home is immaculate.. i say that because if you're going to have animals in your house, even cats and dogs, you have to be willing to do daily cleaning, not once a week cleaning. Its just how it is. But i do that without animals, so the animals are my bonus
I started, but didn't finish an article on house chickens.. ill do that when my house projects are done