Good With Dogs:⬛⬛⬛⬛⬛⬛⬛
In short, yes, the dog can be kept outside as long as the dog has suitable shelter, water and a large fenced in space. However, if your mother is so afraid of dogs that she should not come in contact with the dog. I would not suggest getting a puppy at all. Puppies are boisterous, playful and unpredictable. They can set off fears in people with a fear of dogs already and that energy is read by the puppy and they can become fearful, or even aggressive towards people like your mother/women. Search for a calm, adult dog that is already socialized. This will allow your mother to read the movements/behaviors of the dog from afar and get to know the dog so she can say hi when she is ready. But do not bring a puppy into that environment, it is not healthy or suitable for both your mother and the pup.
The American Foxhounds are good with children, they are loving and loyal family companions. The American Foxhound has a high prey drive, but if socialized correctly and trained then the dog should do fine around small dogs.
Foxes are able to run between 30 and 40 miles per hour at their fastest depending on the breed, and the American Foxhound was bred to be able to run just as fast, if not faster then the fox.
And its not so much how fast but 'for how long'. My Foxhound can maintain around 12 to 15mph for 2 to 4 hours when on trail. Once he has the scent, 'Stand back and let him do his STUFF!"