Yes. By proximity.
Yes. If the toothache is on one or more of the lower anterior teeth.
It's possible. The chin is innervated by the mental nerves which are the final branches of the trigeminal nerve which innervates your lower teeth before exiting to your chin. It is possible for any pathology--toothaches, cysts, cancer--along the path of innervation to cause pain or numbness in the chin. See your dentist.