L.B.
I have cats, dogs, dust mites, trees, and grass - all of which my son is allergic to according to his allergy testing. But, I did not do the official scratch test on him until he was 14 years old. He had asthma all his life, as he hit puberty he began to age out of the asthma but his allergies worsened - so he and I both decided it was time to figure it out.
His response to his doctor, upon learning what he was allergic to was "Well we are not getting rid of the pets and I mow the lawn. What do you suggest." Doctor was a little taken aback, but went on to explain that since he had been exposed to our pets for so long he was most likely desensitized to them and told him he could wear one of those little white masks when he mowed. Yeah, being a teen that was a fashion no. LOL
And so far, he hasn't had any problems above and beyond his normal allergies. Knowing what he was allergic to really only ensured me that I was not missing anything. 5, 6, and 7% allergic to something is not very high. My son was much higher than those numbers, on cats especially, but he can pet them and love on them and not have adverse reactions that his normal allergy medicine will not handle.
What I have learned is that as our children grow their bodies change and they can become desensitized to allergens that they come in contact with on a regular basis - this, in fact, is the basis of allergy shots. Small amounts of exposure that build up resistance to a specific allergen.
Since your child is so young, I would discuss standard allergy medicine remedies that treat the symptoms, and consider re-testing in a few years.
I would not get rid of the pets. What we do is sweep and mop more often and stay on schedule with his allergy meds. If he does have a bad attack he pops a Benadryl, takes a shower (to get the allergens off his skin) and goes about his day.