Chonic Alternator Failure

With a weak battery and high internal resistance, it could be doing that but a cranking drop to 10.5V as reported is not all that bad; most are spec'd to 10V or less under cranking loads as you noted. The symptoms as orignally stated points to diodes blowing, and with the overvoltage output when cold, I would focus on the regulation issues 1st. If the car has been run like that for months (as it seems), that has not done the battery any favors...