If your wife's family has a history of asthma, can it transfer to your kids?

Yes, sort of. Allergy, and in kids asthma is almost always a symptom of allergy, runs in families. Major risk factors for asthma in kids are (1) mom has asthma (2) the child has eczema. If neither parent nor the older kids have allergies or asthma the risk that the next child will have asthma is low regardless of asthma in other relatives. Whatever the risk you can cut it in half by not smoking!
Yes. Much of asthma is genetic so if you have a family history its certainly possible to pass on to off spring.
Yes. Asthma often has a hereditary component. If one parent has allergic asthma, a child has more than 3-4 times greater likelihood of developing asthma. This rises to 6-7 times if both parents have asthma.
Yes. There is a genetic predisposition to develop allergic disease. It may be asthma, rhinitis, eczema or other problems. The incidence of allergy in the general population is about 20%, if one parent is allergic, the incidence is about 40%, and if both parents are allergic, the incidence is about 70%. Obviously, there are many other factors in developing any form of allergic disease.