Thanks again to everyone for the kind and detailed responses. I will try to clarify as much as I can and add an update. The dog is owned by the son of the homeowner. I live in SoCal and the dog owner lives in Nocal. I had seen the dog three or four times over the past six months. I have come to learn that the son has been visiting frequently for business; apparently he brings his dog with him to his mothers house.
From what I understand, the dog jumps the back gate at will. On the day it happened, the son/dog owner, his wife and kids, and his mother left for Canada. As mentioned, they left the dog at the mothers house in care of the neighbor. So in this case, the dog jumped the back gate, which I presume they knew he could, and was in the street when my kids were playing. The exact location of the incident is unclear, as the stories of my son and my daughter differ to a degree. According to my son, it was in the yard. My daughter suggests it was on the sidewalk. Either way, she had free access to the dog.
And yes, I have, on numerous occasions warned my daughter about approaching dogs. Unfortunately, she would have approached a polar bear to pet it if she could. She is very stubborn and as said, seems to have learned a lesson here the hard way. Two other things of note; I am 100% confident that my daughter would never intentionally hurt or scare and a dog. In the other words, she was not beating it with a stick when it bit her. Two, she'd actually interacted with the dog in the past (the first time it got out). So you might say she was even more comfortable with the dog.
As for attorneys, I have spoken to a couple; one off yelp and another referenced to me by a friend. I gave them details, they said they'd be happy to take the case, but I have been noncommittal up to now. As for making claims against three insurance companies, it is my understanding that I cannot make a claim with any company but my own. It is the insured that would have to make them. And I have not gone that route except for with the dog owner.
My daughters physical injuries were moderate I suppose. She had the one significant gash that required six stitches and one puncture/hole. It was extremely scary looking at the time. The stitches looked even scarier, as it did during the first week after the incident. We were very concerned.
Now two weeks out, it doesn't look quite as bad. I think see was lucky from that standpoint because the bigger scar runs across her jaw line. It isn't as visible as it could be and appears to be healing well. As for my daughters psyche, she is at least somewhat affected. She is already sensitive about people looking at the scar, and on her first day back as school asked my wife to apply makeup to hide it. She doesn't want to talk about it at all.
Hopefully this information provides some useful insight.
As to the update, I got a call from the State Farm adjuster this afternoon. It was during work hours, where I do not have much reception, so I missed it. I did not call back because I was not sure what to do. But I figure I will need to figure it out quickly.

