Why does Han Solo's sorry about the mess not have one pip?

Does barrage generator had three suppressive tokens after the attack is resolved if there is at least one damaged ruled?

Dodge and cover resolve before Pierce ever takes effect correct?

1. So that the player using this card gets the intiative if an opponent plays an actual 1 pip card.

2. Two. One, as usual, for rolling at least 1 hit or crit, second from Suppressive granted by the generator.

3. Correct.