Based on my experience, the cat and dog could have more to do with your child's problems than you think, and recommend some testing. (details of my situation below)
My son was diagnosed by a CHOP allergist with a cat allergy a year ago when he was 2 through a skin test (where they use a needle to inject some cat allergen under the skin...no more painful than a regular shot). My husband is allergic cats, so it was no surprise.
The surprising part was that he had no outward symptoms of being allergic to the cat my babysitter had (he goes there every day and would never had sneezing or coughing, etc., at her house). We were actually there to check for a peanut allergy. But before we found out about the cat allergy, what was happening is that he would break out into hives for no reason randomly, even on the weekend after he hadn't been to the babysitter since Friday. Then, he was having a lot of respiratory problems, where he would get colds and they wouldn't go away and I had to give him breathing treatments with albuterol for a week at a time to get the cold to go away.
We found out he was allergic to cats last fall, at which point my babysitter decided to give the cat away, and I have not had to give him any breathing treatments or anything since then (over a year ago). What we were told by the doctor is that he was that his body was constantly fighting the allergens from the cat, so he was more suseptible to viruses (hence, the hives and respiratory problems). So, when he was at the babysitter, even though the cat would hide for most of the day, the allergen from the cat was around the house and it was wearing down my son's immune system.
I hope that helps. I recommend the CHOP office in Exton.