Will exposed bone resulting from a bone graft heal itself?

Yes, but compromised. All wounds will heal in time. However, exposed bone during grafting will compromise the result of the graft. Most often this is due to infection. Must keep the area clean. Consult the dentist/surgeon who do the graft.
Probably. Your body is pretty good at healing most intraoral wounds over time. That said, it really depends on what kind of bonegrafting procedure was done, which material, your medical history, etc. You should seek a local dental professional for an evaluation.
It is possible. It depends on the size of the exposure. It it is small, usually it does heal but itself. Your doctor may place you on a Peridex (chlorhexidine gluconate) mouth rinse. If the exposure is larger, then most of the graft may be lost.