I am not surprised that the poor bird is having trust issues after going through all that.
If your dog tried to eat her, then I think that I would see an avian vet, if it were me. There is little point worrying about her trust, if there could be serious health issues. Dog saliva can make parakeets ill. And there could be injuries that you don't know about.
She must be in shock ~ poor thing.
If she is
ok, health-wise, then she is still going to need a lot of time to get over this ~ and will not want to be anywhere near the cat and the dog.
Do please make sure that neither your cat, nor your dog, can get to her, again, in this way.