Depends. Asthma and allergy experts have been telling parents for decades that having pets (especially cats) at home is not good for children with asthma. That advice is still very true for the asthmatic child who is allergic to pets (>50%). Interestingly, more recent information has shown that if a child is born into a household with 2 or more pets, they are less likely to develop allergies and asthma.
PROBABLY. Sometimes animal danders could effect ones asthma if you are allergic to cats. But if not its ok.
Yes. If the child is not allergic to cats (can be tested) then it is fine to have cats around.