The allergy testing is done either on the back or forearms. First they number the spots, usually 40-100, depending on how much stuff they are testing for. Then they take a special piece of plastic and use it to scratch the skin and put a little drop of the substance being tested on the scratch. It doesn't really hurt, but it's kind of annoying, and by then end you just want them to stop. But it only takes them a few minutes to do all of those because they're really fast. Then you sit there for 10-15 minutes and the things that you are allergic to will turn into welts, like mosquito bites. Then they look and measure the size of any reactions. Usually there are a few reactions that are questionable, so then you end up with 4-10 more specific tests, where they take a needle and inject a little bit just under the skin (kind of like a TB test), and look for a reaction. The nice thing is that when they are all done they wipe it down and rub on some anti-itch lotion that feels really good.
When I started shots, I got two shots twice a week. You can also just go once a week. The way shots work is that you start at a really low dose and work your way up to a higher dose, the maintenance dose. Then you stay at that dose and space out the shots, going every 2-4 weeks for a few years, and in theory at some point you are done. So in order to get up to the maintenance dose higher, I went in twice a week, rather than once a week, at the beginning. Then at the maintenance dose I went every week for a while, then every two weeks. The number of shots depends on how many things you are allergic to.
I think the shots helped me the most in terms of my asthma. Right now I'm not on shots and I'm still doing pretty good. I currently take advair, zyrtec, and rhinocort. If the asthma is caused by an allergy, then an antihistamine or something may help as well. There's the stuff like zyrtec, allegra, claritin, etc. Then there's singulair, which is something different.
The person to see for allergies and asthma is an allergist. Most of them treat asthma as well, since they so often come together. But you would want to verify when choosing a doctor that they do have experience with both allergies and asthma. You might also look for a clinic that specializes in pediatrics, because they might be more child-friendly about the testing and shots.
You are doing the right thing keeping the cat out of his room. Also, vacuum as often as possible. Have him wash his hands after he pets or plays with the cat. Brush the cat frequently to cut down on shedding. It's not actually the fur that causes allergies, but the proteins that are in the cat's saliva (which get deposited on the fur), but getting rid of as much extra fur as possible does help. It might also help to wipe the cat down daily with a damp paper towel.