Osohapi, my daughter is allergic to tumbleweeds too! I know - they're dead, right? It's weird.
Anyway, here's my two cents.
First, if your son's skin prick test shows an allergy to dogs, or other things, it will also show a degree to which your son is affected by a particular allergen. For example, my daughter's skin test showed a fairly mild allergy to cats and dogs, but an extreme, clinically significant allergy to dust mites, spinach, beets, quinoa (all members of the amaranth and chenopod families), and tumbleweeds. So it's possible that your son might have a mild allergy. It might be quite manageable. Don't despair yet.
Second, if your son is allergic to dogs, this could be an important lesson about health. I think that to say "you're medically impacted by something that seems simple and quite normal - a family dog - but we're just going to overlook it", there are implications that could develop down the road. To say "your health is less important than a pet, or an ice cream cone, or working out too long in dangerous heat just to stay on the team" is to endanger our kids. Your son's allergy, if there is one, is not his fault. It's not as though you have to get rid of the dog because he bit your son because your son kept abusing him. I think it's important to teach our kids that their health is important, and sometimes it can mean having to give up something.
Third, be sure that your son has had adequate and complete allergy testing. Besides the skin prick test, there is a blood test for the 26 most common food allergens (corn, dairy, soy, wheat, etc). It's very simple (well, for the patient at least - it's just a blood draw).
There's a more in-depth test called "patch testing". My daughter just had it done. She's already had the blood test and the skin test. But they were looking for more. The patch test involves sticking a series of sticky pads onto the back of the patient. They don't hurt. Each pad has about 6 or 8 little dots on it, infused with substances. The pads are matched up with a grid in a chart so that the allergist knows which dot corresponds with which substance. For 5 days, the patient does not shower or exercise or swim (a basic sponge bath is fine, as long as the back is untouched). The pads are peeled off and the allergist looks for a reaction, such as blisters, extreme redness, etc. The substances that the test looks for are not so much grasses, trees, foods, etc, but metals, common additives in medications and lotions and soaps, dyes, things present in soils, all kinds of things. My daughter was tested for 48 substances. She tested mild (fairly insignificant) to two, but severe (clinically significant) to one, and from that we have identified an important source of one of her medical problems. So you might consider the patch test since it appears your son has allergies that aren't just explainable by the dog.