Okay, so the doctor told you it doesn't matter one way or the other? Really, I can not emphasize enough the value of having pets at a young age. They teach gentleness, responsibility and so many other things. Now, I say this as the owner of three big dogs and used to have a cat. (We couldn't bring him with us when we moved :( ) . Anyway...I grew up allergic to everything..outside and in..dust, molds, grass, trees, weeds, flowers, cats, dogs, cows, (yet not lactose intolerant..just the fur)..and I lived on a dairy farm when this diagnosis came. I've grown out of about half of them with time and tolerance build up. If your son has issues..see what causes the issues..does he let the cats rub his face? Then you need to make sure they don't do that. Dogs and cats are very trainable..you can train them to stay off of things your son is on. He can sit on the floor, or play in the yards with the dogs. Dogs and kids should not sleep together of course..neither should cats. We tie our dogs to their separate beds at night and the kids let them out first thing when they get up. Your son might be too young for that but he can learn their habits eventually. It seems wrong to oust animals from their homes just because something is wrong with us that we refuse to adapt too. Cats don't take a lot of maintenance really. If you want to reduce shedding then maybe ask your hubby to start adapting them to a comb, or maybe twice a year take them to a groomer who will reduce their coats..especially indoor cats do not need as much winter hair as outdoor cats. With the dogs, again..cleanliness is important but dogs keep themselves cleaner than you think. Dirt you can see is dirt that does often come off with time. But..longer coated dogs take more maintenance. If it won't harm the dog temperature wise..reduce the coat..(you can a dog shaved fairly cheaply and many do it this time of year so the new winter coat grows in clean and mat free). But brushing a dog three times a week for a half hour can greatly reduce shedding. If they are heavy shedders add a couple of tablespoons of fish oil to the dog's food and to the cat's food..it will help and make them look very pretty :)
It does sound like you didn't want the cats around as it is..but I assure you they did not MAKE him allergic to anything...he was born with it. They will not give him asthma..he might develop it. Pets all and all are worth the trouble if you love them.
My advice is keep the pets. Your husband loves the cats so let him have his cats..with rules of course...he must clean the litter boxes..scoop daily! (And you must pick up the slack if he doesn't of course but then again..all part of owning pets anyway). Dogs should be trained positively so that changes in rules don't become "stay away from everyone" rules and make sure they are exercised enough through play and walks etc so that training them won't be a huge chore.
I work at a doggy daycare and many of the people that bring dogs in have issues with their pets of one kind or another. One last thing..don't go trying to bathe the cats and dog once a week or anything..it will give them sensitive skin which will cause itching etc. Dogs don't need bathing but every four to six weeks unless they live somewhere where they get very muddy and cats rarely NEED bathing but a good grooming once or twice a year never hurt them. Also remember if you groom a cat..never cut the hair on the face as you can easily cut a whisker which can be very painful for some. Good luck..main thing is just help him learn not to touch his face and help the pets learn to not touch his face. That's the best way to keep allergens out of his face. That's what I did with cats and dogs..and now I regularly have one or the other curled up near my ear on daily basis. I take meds for my other allergies..I out grew most of my animal allergies.