Components calculations for this circuit. Is my reasoning right?
Below 12V, D1 and Q1 have to be ON. Q2 and Q3 should be OFF. But here
  is where I'm stuck. Does this mean Q1 base is being pulled up through
  R3 and D2?


Yes. That means D2 will glow weakly as it supplies Base current to Q1. Current will be about 0.3mA, so depending on LED efficiency it might be weak enough to not worry about. 

If it is a problem then put another resistor across D2, high enough to not draw too much current when Q2 is on, but low enough to reduce voltage below the turn-on voltage of D2 when Q2 is off. eg. 3.3k at 0.3mA will drop 1V keeping the LED off, while at 2V (voltage of green LED when on) it will only draw 0.6mA away from the LED.