What breeds of dogs can a person with dog allergies own?

No dog is completely hypoallergenic, not even a Poodle which many people claim. Some people do better with certain breeds however. This is a tough question to answer however because we don't know exactly what you are allergic to. Is it the daner, the hair, skin flakes, or maybe even a skin condition the dog has.

Dogs which are "allergy friendly" are those that don't have much dander. Many people may tell you that dogs that shed very little are the low dander breeds but Chihuahas shed a lot however they are low dander so shedding isn't the only issue to look at.

A good friend of mine has two Chihuahas and they don't bother my allergies whatsoever as long as they don't touch my face while I'm hugging them and cuddling with them.

This list below is a list of breeds which are considered "low dander." Dander is what causes your allergies, not the fur or hair.

Airedale

Australian Terrier
Bichon Frise
Bouvier des Flandres
Cairn Terrier
Chihuahua
Chinese Crested
Giant Schnauzer
Irish Water Spaniel
Italian Greyhound
Maltese
Miniature Schnauzer
Norfolk Terrier
Norwich Terrier
Poodle (all types)
Portuguese Water Dog
Puli
Schnauzer
Soft-Coated Wheaten Terrier
Tibetan Terrier
Welsh Terrier
Xoloitzcuintli
Yorkshire Terrier

Some people are only allergic to SOME breeds of dog (my mom can't be around German Shepherd at ALL of she breaks out in hives). I suggest researching the breeds on the list above to find out what their personalities are like. Once you've narrowed it down to some you would be interested in owning, call some reputable breeders near you and ask if your mom can spend some time with their dogs. If she's allergic to the breed, I'm sure she'll find out shortly thereafter. ;)

As far as good with other pets goes, you want to avoid terriers. It is true, like you said, that the animals behavior is how they're raised... but terriers were bred to instinctively hunt small animals. My mom has a Yorkie/Maltese mix and he chases down our cat and chews on her... it's not aggressive... but it isn't pleasant either.

Of the breeds above, the ones that have a reputation for being good with other pets (as far as I'm aware, others may be good too) are as follows:
Bichon Frise
Chihuahua
Italian Greyhound
Maltese
Poodle

Take a look at some of the dogs on the list and figure out if one of those breeds could possibly be the right dog for your family including the allergy sufferers.

Check this out, you can figure out a good breed for your family... it includes pet friendly and better for allergy sufferers.

Good luck! Let us know what you decide!