Yes can move before or after using the window. This is true whether you move one or two floors. An intervening droid on the 2nd floor does NOT prevent movement between the first and third floors. You cannot move or attack when transferring droids to the palace, deploying droids to the battlefield, or recovering damage points. I'm kicking myself for missing this one.
Cards supercede the rules.
It is the 6 unselected action cards. You must discard all or none when this card is played.