Should I sue because of the accident?
Personal injury is an injury to a person. The commercial driver's insurance should take care of the property damage. Unless they are unwilling to pay for the damage, there's no need for an attorney.
Should I file a malpractice personal injury lawsuit?
Why were you going to a chiropractor if you didn't know you had a bad back? How do you know the chiropractor was the one who caused your back problems? How did you strain your back? These are some factors that would be taken into consideration.
Are you responsible for damages if an accident happened on private property?
The lady who hit you is responsible for the property damage. It makes no difference that it happened on private property. If she doesn't have insurance and is unable to pay, check with your own insurance company to see if you have property damage coverage. Your own insurance company will then pay for the damage.
Disclaimer: The responses above do not form an attorney-client relationship, nor are they intended to be anything other than the educated opinions of the author. They should not be relied upon as legal advice.