I've never ran over someone's pet so I've never actually had to put too much thought into this, and I'm very sorry for the lose of your family's pet.
I just have to add this in as a thought, and I was curious what other people thought.
I was always taught to never stop at the houses near by if you have hit a dog or cat. The concern being that the owner may take out their anger on you and hurt you or worse, especially if you are a woman and it's at night. I've always been a bit torn on this topic. Not everyone lives in a safe area, but then again almost everyone loves their pets and doesn't want to think they've hurt a beloved animal.
Overall the best answer is to be careful when you are driving and to always stay alert to avoid hitting anything, people or animals. However, if it were to happen, what would be the best way to handle this situation without putting yourself in a potentially dangerous place?