Is the dentist covered for mistakes?

Remaking crowns hurts all parties involved (the patient, the lab, the dentist).  Clearly it is best to do it right from the beginning, but sometimes "things" happen.  In that case, sound business principles are in play.  Insurance won't pay again (often there is a 5 year wait between benefits, no matter the reason), but the lab often remakes at no charge if remade within a short amount of time.  There is no "legal requirement" for any of this, so in theory a dentist could charge a patient for each remake, even if done 5 times in two weeks.  That is not good business practice and the dentist likely would go out of business if routinely doing so.

Your dentist will not be allowed to rebill your dental insurance for a crown that wasn't properly fitting or otherwise unacceptable.  If the crown doesn't fit properly, and sometimes this isn't discovered at the time of placement, then most dentists and labs will redo the crown at no charge.  I doubt that your dentist will try to bill either you or your dental insurance for redoing the crown.