A police report isn't going to work when your wife willingly gave up the dog.
I really don't mean to offend but I don't want to beat around the bush, so I'll say it bluntly. You and your wife have issues. Your wife decided to give away one of your family dogs behind your back, after you voiced your disapproval, fully knowing her own kids love the dog. She didn't take you or your kids into account. She listened and sided with the group of people who absolutely don't have her interests in mind, as opposed to her own family. And this family has a history of walking all over you and your family for the sake of their own personal gain.
This isn't really about the dog. Your wife doesn't have your back, which IMO is one of the most basic concepts of marriage. You need to really think hard about just how committed she is to you and your kids, because right now from the incidents you've been describing it doesn't look like she's committed at all. This sounds like someone who folds like a cheap tent for her family who are walking all over her, and she doesn't care if she's dragging down the rest of her own family with her.