California is a strict liability state. So, yes, you do have a case.
You can read up on California's dog bite law at:
http://dogbitelaw.com/statutory-stri...a-dog-bite-law
From a liability standpoint you have three possibilities: the owner, the neighbor at whose house the dog was, and the other neighbor who was supposedly taking care of the dog. Any or all of them could have all or part of the liability.
Whether or not you get a lawyer there is nothing neighborly about this. When you accuse people of negligence and demand money from them (that's what you are doing, so don't sugarcoat it) they become your enemies and will treat you as an enemy.
The proper way to do this is to put all three on written notice of your child's injuries, their negligence, and that you will be looking to them to pay damages as soon as those damages can be quantified.
Those letters should elicit responses from insurance companies that will represent their policyholders and either defend their policyholders or enter settlement negotiations with you.
Up to you and depends on a lot of factors which include but might not be limited to the cost of medical treatment, the extent of any facial scarring, length of recovery, emotional trauma to the child.
If you'd like to reveal medical costs to date and explain the other factors mentioned I may be able to provide helpful comments.
Which should be followed up in writing to the neighbor, the owner, and the other neighbor.
Could be either, could be both. You'll find out when one or more insurance companies respond.
What generally happens is that you don't get any money from anybody until the claim is ready for final settlement, which could be months away since you want to make sure that your child is completely recovered and monitor the scarring for a few months to see if that clears up or not.
Meantime, your own medical insurance covers it and your medical insurance company is likely to require reimbursement out of any settlement that you get on behalf of your child.
Two weeks with no contact from any insurance company? I strongly suggest you get those letters out. Then, if you still don't get contacted by anybody's insurance you can decide whether to sue in small claims court if the $10,000 limit takes care of it or hire an attorney if you think the case is worth more than that.
You can get small claims information at:
http://www.courts.ca.gov/selfhelp-smallclaims.htm
Let me know if you have any more questions.

