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As Antimatter said, you should look into behavioral training. Maybe your vet will know of a good place. Hope everything works out.

Sorry to hear that, feel well soon :c

I think the dog had a grudge against the cat, but i can understand your worry over your kids.
I have had to get a rid of a dog i had before my kids, she started growling at my son for no reason at all, so i got rid of her to a shelter, was a sad day but i didn't feel i could trust her, even with more training, she needed to be in a home without children.
My other dog has never been aggressive so we happily kept her.

Not sure what to advise, i'm tempted to say wait until the dog shows any sign of aggression towards humans, but by then it might be your dog shows it's first time aggression by biting them.

Try the training, or just chat with an expert vet or trainer and ask them how you know whether you dog is safe around people now.

sorry to hear about your cat also.