Can a herniated disk heal itself after 3 cortisone shots without surgery?

Maybe. Cortisone injections (why 3?...There is no standard to do 3 shots.) can help the inflammatory process of the nerve roots irritated by the disc hernitaion. These injections do not "heal" the disc, but can cause accelerated pain relief. Disc herniations occasionally resorb (disintegrate) over time, but there is no guarantee that a disc is healed once the pain is gone, even from injection therapy.
Sometimes. The purpose of the cortisone injections are to decrease swelling and inflammation around the disc. While the herniation may not be healed, the pain and disability definitely can be, that may allow for more effective pt which may allow the herniation to improve.