Is the property responsible for the damage to the door of the residential area?

No fault, no responsibility. According to the property contract you signed. If the property fails to fulfill its safety management responsibilities, such as whether the patrol is in place, whether the duty is 24 hours, whether the monitoring equipment is turned on, etc. If there is a fault, considering that the behavior of the property management company is not the direct cause of the damage, and the ability to prevent accidents is limited, you can request compensation appropriately.