If you filed on your own insurance and it paid for the property damage then your insurance company may or may not go to the at fault insurance company for payment. The insurance companies have agreements with each to work out these disputes. Since the property has been satisfied by your company the at fault company will have no responsibility to you for the property damage. This does not however, eliminate your option to seek other damages such as for injuries. And it should not interfere...
Obviously every case is unique. Damages in every case are also unique. They depend on medical billings, lost wages, other out of pocket expenses and also non-economic damages such as pain and suffering, etc. You should have a personal injury attorney evaluate your case. Do not visit with the insurance company without having done so first and certainly prior to signing anything from the insurance company.
The answer is most likely. You have to review all safety equipment on site. State, city and local statutes and ordinances for violations. And many other variables. They are complex cases but can be successful. I have recently tried and settled a similar case in Houston for a apartment complex drowning. The city inspections and code violations help with that case. You should contact a personal injury attorney who have experience with these types of cases.
If you have uninsured motorists or personal injury protection insurance coverage then you should definitely call your insurance agent. You might also consider calling a products liability attorney to take a look at the vehicle, if the vehicle is still available. Other than that you should call an attorney to at the minimum and get some advice on what other options are available to you.
If this case happened in Texas then the statute of limitations is two years from the date of occurrence. You should determine the exact date of injury and contact a personal injury attorney as soon as possible. If it is not too late you have then have them file a lawsuit to preserve your claim.
As usual your child's health should be of first priority. The owner of the dog or even the store can have some negligent exposure for the dog. Typically, they dogs have to have some sort of history of behavior problems but it isn't always necessary to prove up a case. My office has handled several of these type of cases, but you for sure should consult with an attorney before talking to the store or the owner of the dog.
You need to call a Texas attorney ASAP who can look at other possible areas where recovery might be made. For example a product defect case against the vehicle manufacturer. But these type cases need to be worked up quickly as evidence can disappear or be destroyed and reduce your chances of recovering. I would be happy to take a look at it.
Hopefully you got the vehicle plate number and a name and call the police and report the accident as soon as possible and give them all them information you have. I would treat it as a hit and run if I was you.