Many people have allergic reactions to dander and hair. Cats are a particular problem because they lick themselves. So, the hair also contains saliva, and both can trigger reactions. If this poses a problem, it might be best to keep your pets outdoors, if possible. At a minimum, keep them out of your bedroom and carpeted indoor areas.
Feathers are also problem.
They can trigger allergic reactions even if they are contained within pillows or mattresses. Mold flourishes in many dark, humid household places. It is often found in showers and basements. Mold is also found in other places such in plants, kitchens and other wet spots. Mold should be removed to the extent possible.