I grew up with cat allergy and a cat. I was such a cat lover that I would rather suffer then let him go.
Back then nobody did nothing special to reduce my allergy. I had daily swallenn eyes and runny nose.
Nowdays t here are ways to keep it low and here are some tips I found for you:
http://www.ehow.com/about_7238436_spray-cat-allergies.html
seem that if you just google " cat allergy relief "
and "cat allergy spray relief"
you can get tons of advices and info what works best.
I would keep a cat just as most people suggest if the allergy is not causing asthma because asthma is very dangeorus.
Also keep in mind that the allergy for any specific allergen might be higher if a child is expsed at the same time to other allergens.. as in censory overload?..
simply if your child is allergic to dust and body is exposed to lots of dust, if the cats walk in, the system is so overloaded it reacts.
Now as far as environtment I would put my mony on the following first and foremost:
* I would remove all carpets made of or containing olefin, polyester, and other nylon and non natural carpets. ALL OF THEM. I would put wood flooring everywhere.
* vacumming? people say.. beware, vacuming actually on one hand helps allergy but on theother it sends TONS of dender and small particles into the air so as end result person might suffer more if you vacum more..
so vacumming is good thing as to removal of debris but it has to be done SMARTLY..
smart vacuming means: NEVER vacum your home when a child is in the home. do it way before her coming so the dust can settle. To do simple experiment what the facum does to home air.. turn off all lights at night makeit pithc black room, vacum a bit dusty carpet and turn on the flash light and see all that flights in the air.
* so.. smart vacuming more.. on the topic: use vacum cleaners that are VERY low emitting ones.. there are vacums that actually do not use suction as a medium .. there are things like Shark and other that only have two brushes inside and they just rollt he brushes inside and collect the dust.
Or.. use big Guns vacum once weekly and on daily basis the low emitting roller vacums they are usually the ones who you have to charge.. as they dont' have srrenght to blow anything.
* smartest thing is not to vacum at all but instead do wet cleaing..
wahing floor, wipping floors.. steam cleaing.. WATER ONLY steam cleaner.
NEXT THING MOST IMPORTANT THROW OUT EVERYTHIGNT CHEMICAL BASED OR AROMA BASED THAT EMITS INTO THE AIR SMELLS AND CHEMICALS AND HEAVY METALS AND THE LIST IS VERY LONG BUT SIMPLIFIED WOULD BE SOMTHING LIKE THIS:....
* First dump all Air refreshners, and NEVER look back. They usually contain bad things and do read about them and it is hair raising, also there is no legal regulations what may and not go into them so there are even heavy metals that are emitted into the air.
* next go all home cleaners.. ALL!
all you need to clean yoru home is Baking Soda, and Vinegar!!!
amazingly enoguh read green home pages and you will see that people are successfuly for years using those two to clean EVERYTHING.
commercial chemicasl are what they are because nobody can cash on vinegar and soda so it is cheap for you and best for your kid and home and your pocket and everybody will love you for that. Vinegar smell goes away after half hour or less leaving home fresh smelling as it actually KILLS bacterias and odours not COVERs them.
* Thorw out everything Perfume. All perfumes are killers for kids and people with allergy and many does not even know, the same about headaches sufferers. Just read more about dangers of prefumes.. even for pregnant ladies.. again hair reising.
* Buy only things without perfumes.. : soaps, shampoo, washing powder, detergetnts.. i think Seven Generation and other companies do carry them.
This eliminates greatly daily emission of smells and chemicasl into your home that actually has nowhere to go but into your lungs.
* NEVER NEVER NEVER use any of the perfumed sheets for the dryers. they are major complaint for allergy suferers, and they emitt so much to your home
* Diet - don't feed your child things taht contain bad things .. additives, conservants and artificial colors as body has to fight them off so this make system works harder..
* Your child should eat lots of raw diet, raw veggies and drink smoothie without icecream.. just fruits and water.. fresh.. daily as this works as a bad stuff magnet.. jar jjuices are dead juices.. fresh smoothie is alive.. it simply binds with bed chemicals from the body and removes them
* You might look into sprays for the home but try to look for natural ones not chemical based as you might be adding more to environement.
* NEVER use any candles.. cancles are bad even without smells..they shoot lots of bad stuff into your air, rad about dangers of candles and cancles and allergies.. and never never never use perfumed candles.
* use perfume free kitty litter as this alone is really bad for the home if someone is using perfumed litter, bad for a cat, bad for allergies in people.
wash your bedding often.
*Most imprtantly let lots of fresh air in the house. Check when the air is least polluted as probably eavenning.. this allow all bad stuff to go out and new air get inside. do it daily .
LASTLY.. AND MOST SURPISINGLY WHATMANY PEOPLE DOES NOT KNOW
DO YOU KNOW WHAT IS BEST HELP FOR ALLERGY?...
ACUPUNCTURE


tHAT IS RIGHT, YOU WOULD NOT BELIEVE IT..
Yet read more , resarch more and you will be amazed.
Someone in my family suffered from heavy allergy that was simply and easily cured with Acupuncture.
Acupuncture rewires the system.. amazing.
P.S. if you decide to clean the ducts.. do not desinfect them as this will bring more chemicals into the house and do read all about possible problems from duct cleaniing
and how to do it right.. there ar better ways and not so great ways.. but genearlly it might be a good thing to do.. but just cleaning by vacuming them not
by sanitizing them.
I would think that washing a cat is really a minor thing.. if you do all other well.. and also brushing of the cat actually sends more dender into the air .. unless you would brushvacum your cat but this is very difficult to do as they are afraid of the noise..
oh and btw.. if you are sitting on millions.. then best vacum of all in the world is a vacum that you can install in the garage and you ahve just outlets for it all over the house.. and you simply connect the hose and vacum anywhyere you want. This is best because it does not shoot dust into the air but only sucks in. this vacum cost like 1500 to 3000 installed.. and I think it is called Central Vacum system for the home or something. I want one bad but don't have this kind of change under my sofa

Hope this helps.
Best regards.