[papajam]

To answer your last question first, yes, you can drive the car without the regulator BUT only 'til the battery goes dead. Depending on the battery's state of charge/condition and the load it's under, that could be anywhere from 5 mins. to 5 hours.
The reason to check the voltage with the reg disconnected is to determine if the alt has an internal short from battery to the field terminal. Since the alt is not full fielding with the reg disconnected, it would appear that the alt is OK.
The second green wire on your reg plug goes to alt light on the dash. The pic was taken from a 68 Euro version which has the second green wire attached to the alt instead of the plug.
Well, it would seem that regulator is the problem. But 3 shorted regulators? Possible. Were any of them known to be good before installation? Were they properly grounded?
My thought would be to take the alt back to the rebuilder and have them benchtest it with regulation. If it tests OK, I'd install a new regulator.