Can spinal disc problems ever heal on their own?

Absolutely. Almost 40% of the population by statistic have a ruptured disk in their back without symptoms. It happens at some point in time but they do not keep their symptoms. Of course, sometimes the herniation is so severe that surgery is necessary to remove it.
Yes. Many can get better with time and conservative care within 6-8 weeks. Check out Spine-health.com.
Yes most do. Herniated discs occur frequently on mris. Many are not painful. If there is a painful disc on an mri, the symptoms will resolve more than 90% of the time. This is not to say that the herniated disc will go away. On the contrary, the disc will often not appreciably change. Pain from a disc typically needs both nerve compression and inflammation. Resolution of inflamation leads to pain control.