If I got into a car accident in my mother's car but she had no car insurance, am I responsible for all of it?

Your mother is liable for the accident because she is the registered owner of the car you were driving.  If the other driver had uninsured motorist coverage on his/her auto insurance policy, he/she could file a claim through his/her insurance company.  The insurance company would then seek to recover the amount it paid on the claim and that is why you received that bill for $3500.  Since your mother didn't have insurance, she and you could be named as defendants in a lawsuit to recover the property damage to the other vehicle and if applicable compensation for the personal injury claim of the other driver or occupants of the other vehicle.  The personal injury claim would include the medical bills, compensation for pain and suffering which is an amount in addition to the medical bills and if applicable documentation of any wage loss.  Compensation for the medical bills is straight reimbursement.  Compensation for wage loss is straight reimbursement.  The medical reports will document the nature and extent of the injuries and will be used to determine compensation for pain and suffering.

If you and your mother are sued for negligence and the other party gets a judgment against you and your mother which you can't afford to pay, filing bankruptcy might be an option to consider.  If there is a judgment against your mother as the registered owner of the vehicle, she could sue you to recover the amount of the judgment against her although that is probably unlikely to occur.