I have a DS with ASD, Asthma and eczema.
I have three very small rescue dogs.
Two are long haired.
Their mixes are difficult to determine but one is chix pom, one is allegedly x peke and the other looks like a tiny GSD x Husky (but she isn't).
The short hair sheds the most. Possibly because she is (long) descended from working dogs.
I bath them once a month and I keep their coats clipped fairly short. They don't need long coats. We live in a centrally heated house in the city.
They are not allowed in the DC's bedrooms.
I hoover and damp dust a LOT.
Dogs and kids with ASD can work well but you need to be realistic. If your DC has trouble reading cues in humans its not automatic that he will be fantastic at reading them in animals.
We have to explain to DS what growling etc means or he wouldn't know!
He can take it very personally if the dogs 'don't like him'. He means if they don't come when he calls them etc.
I got little dogs to fit in with our household but don't assume little dogs = easier than a big dog.
Little dogs can be very vocal (is your DS bothered by noise?), far more barky than any larger dog I have known.
You will find your dog and it will be fine. As long as you put plenty of thought into it.
I agree with Lem. Don't go for a tiny puppy even though it seems like the best idea. Go for a young dog with a foster background. A good foster would have helped with socialisation and house training.
Good luck. I love my little dogs. I often wonder why I ended up with three though 