I recently went to look for a cat at the pound and
fell in love with a 1 year old female. The thing
is that she has three legs (missing the back right
one). What I want to know is if a three-legged cat will
have health problems such as hip dysplasia later
There is no reason not to adopt a three-legged pet. It is not uncommon for a cat or dog to lose a leg due to disease or injury. These pets typically lead normal, healthy lives. In fact, animals who are missing a leg usually don’t seem to notice it. I have seen three-legged dogs leading the pack at the dog park.
In the wild, an animal who loses use of a leg is at a disadvantage in capturing prey and escaping threats. Fortunately, your cat won’t need to hunt for food, and I doubt that very many threats to her safety will be present in your house.
Regarding your question about future problems: they are unlikely to occur. When one leg is missing, slightly more strain is put on the remainig limbs. However, in my experience this is unlikely to lead to arthritis, hip dysplasia or other major complications.
Once you adopt your cat, keep her inside. If she goes out, she may have difficulty avoiding dogs, cars, wildlife, and other cats. But as long as she stays inside, she should be fine.