My mom has Asthma, and I was born with it. I still haven't grown out of it. It's not a killer disease, so don't worry about that haha. But the probability of your child having it is a 50/50 chance if...
If they are allergic to dogs, no but if they can live with dogs that can shed, it is fine. I do not recommend it to children with asthma since puggles shed and if you don't know if they are allergic...
This is a bit difficult to answer because we don't know about the child's type of asthma.
Is s/he has allergies, then dogs are tricky. I s/he isn't allergic to dogs then the answer is a bit...
No, you are not due any certain benefits because your child has asthma. In fact, you don't deserve a hand-out nor will you receive a dime by exploiting your child's sickness for monetary gain.