didn't Lucy kill the dog?

No, it lives.

Show too much kindness, people won’t fear you. If they don’t fear you, they don’t follow you.

Nope, its presence upset her, because she was reminded of the little puppy she had had as a little girl. It was stated by one of the scientists that the only species the mutant girls had ever been seen to kill were humans. So the puppy had nothing to fear from her (other than a scowl).

I think she just cut the dog loose. The scene seems deliberately made in a way that suggests to the viewer that the dog was killed - later we see Mayu scream and call the dog's name and fear the worst, but then he comes running back to her, alive and well.