What types/breeds of dogs or cats would be the best option for someone with allergies and asthma?

None really. If you have a specific allergy to dogs and/or cats, the best advice is not to have one. There is a misconception that dogs with hair that don't shed are "hypo-allergenic; however, the allergen comes from the animal's salivary glands. The pets clean themselves by licking and the allergen gets on there hair, fur, & skin. This is also why a reaction is worse when the pet licks you.
None. Cats are more problematic in that allergen is contained in dander, hair, saliva, and urine in high amounts. Dogs differ somewhat also in the fact that the breed of dog determines the amount of shedding and dander. In general work dogs like shepherds, labs, etc. Tend to dander more in order to keep their skin healthy. Other non-working breeds dander less.