I would not assume that it is the cats first. I am an allergy sufferer who has 3 cats. First, look in your furnace vents. If you live in an older home, they are probably ALOT dirtier than you ever would guess, and when your air and furnace run, it blows all that crud around. If they are dirty, clean them, also, change your filter once a month. Another suggestion is to keep the litter box clean and don't let your cats go outside. Outdoor cats get all sorts of allergens on their fur, and when you pet them, on you! I break out in hives when I pet an outdoor cat, but I can cuddle with my indoor cats all the time!
I still think a possibility for your son's difficulties could be teething. Teething is no fun for alot of children. They get all the symptoms you have described; coughing runny nose, fever, infections, etc. He could have gotten a secondary infection caused by all that. I would be taking him back to the MD and making sure he does not have an infection. If he doesn't, go to iherb.com and order the Viburcol. It is a homeopathic remedy that works wonders for baby's immune systems. We have referred lots of famillies to it and it works very well. It is premeasured liquid that does not taste bad. ( I tasted it myself to see).