When a device is activated, and its arm moves out of the 60 cm per side cube, does the entire device still have to be less than 60 cm in height?

Yes the arm may leave dimension during the shot, but cannot violate rule 5c. No part of the launch device may extend outside of the launch area before or after a shot. If part of the launching device extends beyond the launch area during the launching action, it must return to and remain in the launch area immediately after the launch without assistance of the competitors.