I am not sure which insurance company is giving you money for your car - yours or the other person's? In any event, you can settle the property damage aspect of your claim without losing your right to pursue a claim for injuries. You just want to make sure that any agreement or release you sign only releases them for the property damage and not everything else.
Every case has to answer the questions of liability and damages. I would guess that the insurance company is not arguing your injury did not occur as a result of the fall, although it may be arguing the severity of your continuing pain and suffering. Usually with low offers such as yours, it comes down to liability. Essentially, the insurance company is questioning your attorney's theory of negligence or wrongdoing on the part of the apartment owner and/or manager. It sounds like they don't...
You are expected to drive reasonably under all conditions. Reasonableness varies depending on the individual situation. It is the factfinder's job to determine whether the circumstances here are reasonable or not. Without more facts, it is a bit difficult to answer. However, my gut says that even with children yelling, one should not back up like this. Clearly, this action was not done on purpose, but the driver may still be held accountable.
Unless you can show the town was negligent or committed some wrongdoing, it is unlikely you could recover from the town. As a practical answer, you would have to sue the town in Superior Court and this would most likely cost your Aunt more money than the deductible she is trying to collect.
It is really impossible to answer your question because it does not provide enough information. Are there issues with the case? Is the insurance company denying? Are you treating for the accident? My best suggestion, if you haven't done so already, is to contact an experienced personal injury attorney in your area. He or she will ask the pertinent questions to get you the answer you are looking for. Best of luck.
You should have waited to get the keys from the landlord. Not having the keys did not mean you should have created the risk you did. Your actions were not a foreseeable event by the landlord and he is therefore not liable. Your climbing the roof was your own decision, and you are responsible for the fall. Hope you recover soon.