There are many breeds of dogs that are hypoallergenic and should be safe around asthmatics. As for outgrowing the asthma I'm not sure, my son was diagnosed at like 2 or 3 with asthma and took treatments with a nebulizer and albuterol but he's 17 now and has no asthma for many years now, I guess it's possible to outgrow.
Dogs that "don't shed" still loose the dander, it just remains in the coat and doesn't float in the air. They will need to be groomed regularly to get rid of built up dander and loose hair. If this is done, theses breeds should be safer around allergies, safer for the people that is.