Cataholic our youngest son was born with allergies. The first time he started sniffling we thought it was a cold, but he was clear and the pediatrician said it sounded like allergies. By the time he was one he was on allergy meds and he didn't go through the tests for anything since he could flare up at anything and everything. As a teen we thought he was outgrowing the allergies but by 18 they returned with a vengeance. He now has to carry meds everywhere he goes as he could eat something that he has eaten before without any effects but suddenly be gasping for air on a particular day. And yes, his eyes are the most annoying part , he constantly holds a cold wet facecloth on his eyes.
As a child he had a dog, we never deprived him of this as he was dog crazy. Our doctor didn't think it would make much difference for the allergies anyway.
His allergies are year ronund off and on, so I don't think there is a specific time for this.
Sorry to hear about your son, hope it's something else but I think they learn to live with it as they age.