If a 10-year old's skin test ranked as a class 3 for dog is it possible to ever get a dog if you follow proper precautions?

There are various hypo-allergenic dogs on the market right now. A Bischon Frise, I think that is the spelling. That is an example of one. It is a small dog that resembles a poodle. That wouldn't have an adverse reaction with an allergy.
Don't despair, kids often grow out of allergies. When I was little I was deemed "allergic" to every kind of hair; dog, cat, horse, sheep, you name it, plus about 200 different kinds of plants. By the time I was an adolescent I was showing no symptoms and we had cats, dogs and wool blankets in our house.
There is no such thing as a hypo-allergenic dog. It is the dander that causes the allergy and all dogs are going to have dander. The breeds that don't shed may be a better alternative then a shedding dog as they are said to produce less dander.

Yes, all dogs have dander, but there is such a thing as a hypoallergenic dog. It's when dogs do not have as much dander, therefore the effects are severely lessened.

You can most certainly get a dog if it is a breed that sheds less dander, like the one above. Just Google "hypoallergenic dogs" and you'll get some lists. :)