Thanks for the additional information. It sounds like she has a problem with redirected aggression. What this means is your dog gets very excited and possibly even aggressive toward these other dogs and cats and when they are unable to get to them, they redirect their aggression toward the nearest object which happens to be you. It is also possible that in the past, this jump up and bite caused the owner to let go of the leash and the dog got what they wanted as a result.
The only real solution is to desensitize the dog to the stimulus (cat and other dog). `I'm going to give you may method of doing that and then give you another site that uses another method of dealing with dogs that are aggressive toward other animals. Then I have another tool that may help with the situation. I do want you to work with your dog in the muzzle so you are not bit again.
It will be helpful if you can find someone with a cat to help you once you have your dog listening to commands consistently. So you will need to work with her on obedience training. The following site is helpful for teaching you how to train your dog. Be sure and click on the link to the page on obedience at the bottom. and links on subsequent pages leading to detailed instructions.
You want her listening to your commands the first time every time. She should know sit, down, a release command, and come at a minimum. I would also train for the leave it command as well. Training works best if you train at least 30 minutes a day (two 15 minute sessions). I would start making your dog work via the Nothing in life is free program (NILF). It is outlined below.
If you feel your dog already listens to commands well, you can start this step now. What you will do is have your dog on the leash. You will have your helper off in the distance. Your helper will gradually move their animal a bit closer to you preferably walking past your position in the distance. As long as your dog ignores them, you can give your dog praise and a treat. Most basket style muzzles allow you to give a treat through the openings. The second you see her fixate on the other animal whether you are using a cat or dog or if she shows any other sign of aggression (hair standing up, etc.) give your dog a correction by giving a short tug and a firm low toned "NO". It shouldn't take your dog long to realize you will not tolerate the aggression and that if she ignores the other dog, she gets treats. Once this happens you can repeat the training moving the other animal closer until she is no longer trying to lunge at other dogs. You will need to practice this when you and your dog are walking as well. Use high value treats like hot dog slivers or liver slivers.
If there is a dog she reacts to, you might do that one first and then the cat since it will be easier to walk the dog off, where the owner will need to carry the cat or have it on a leash or in a carrier.
A method many use for aggression is the BAT method which you can read about here:
Now if she only bites when she jumps, stopping the jump may be enough to stop the progression to the bite. So the obedience training may help keep her in a down or sit position. If she isn't jumping then she can't bite your arm. So definitely do the obedience training.
They make a no jump harness which might be a good tool to use as well. They physically stop a dog from jumping. See one here:
In addition, if the situation is not improving using the techniques I've described, you may have to consult a professional behaviorist. You can usually find a behaviorist by asking your Vet for a recommendation or you may be able to find one using the following site.
I hope this information is helpful to you and you are satisfied with my response. If you would like any additional information or have more questions please don’t hesitate to press the reply to expert or continue conversation button so I can address any issues you still have .