I am sorry to hear of these problems you are having with your pit mix. My name is Rebecca. I am a veterinarian and will do my best to help.
When dogs kill cats, it usually continues. There can be many reasons for your pit mix to have killed a cat and then fought with your dad's Chihuahua. Many pitties and pit mixes are sweet as can be with people but can't be trusted around small dogs or cats, period. They can be pets, but they have to be kept away from cats and small dogs, or wear a basket muzzle when around other dogs. However, that can be a dangerous situation because even with a muzzle, she might send body language threatening a fight and that might cause other dogs to attack her and she would not be able to fight back.
Trying to train her to be safe with cats is not impossible, but difficult, time consuming, and you would have to work with an experienced trainer. Even working with a trainer, may be unsuccessfull. It may be like trying to train a cat not to hunt mice; it is in their genetic nature.
Trying to have her be safe with other dogs is also difficult. When dogs get to be her age, they are just reaching their social and emotional maturity, and 2 to 3 years of age is often when we see female dogs start to fight with others in the "pack" she lives with. It is so hard; most of the time these dogs will sleep with, play with, and seem best of friends with the other dogs. Then a dog just brushes against her when she is sleeping, or another dog tries to go through a doorway in front of her, or a piece of Kleenex on the floor starts a fight. Sadly, these fights usually get worse with time.
In my 30 years experience, the only safe way to own a dog like this, and people do it, is to keep the dog away from the other dog, always. I sure wish I had a quick easy answer for you, but behavior problems are usually complex and take a lot of training, time, money, patience and attention to change.
Please let me know what other questions you have.
A pit rescue group may be willing to take her and place her in a home where there are no other dogs or cats.