The Millennium Falcon can have an extra Pilot.
If one of your Characters would be damaged and discarded from any arena or your build zone, you may prevent that damage and retreat that Character instead. If you do, the Millennium Falcon doesn't untap during your next untap step.
If your Characters can't retreat, the Falcon will not stay tapped.
You can use the Millennium Falcon (F)'s ability whenever one of your Characters would be damaged and discarded. While the "does not untap" clauses do stack up, one turn of not untapping satisfies them all.
Attack dice rolled secretly can still be prevented by the Millennium Falcon (F)'s static damage prevention ability, due to the time when its effect is allowed to occur. (You simply don't know the result until the dice roll is revealed.) Declare your intention to use it during the damage prevention POP, and then retreat the Character if you find out after the dice are revealed if it would be discarded.
You can't use the Falcon's ability when Painful Reckoning is in play if the damage done before Painful Reckoning is applied would not cause the unit to be discarded. The Falcon (F)'s ability is a static replacement effect. It happens at the end of the Play or Pass damage prevention step (i.e., everyone is done playing damage prevention effects). It will only kick in if the incoming damage is greater than the unit's remaining health. So in this case, with only 3 incoming damage, the Falcon's effect cannot be invoked. Painful Reckoning is also a replacement effect. However, it replaces the actual placement of damage counters╤in this case, replacing the placement of 3 damage counters with the placement of the original 3, plus 3 extra counters, or 6 counters total.