If adhesive is used to grasp a ping pong ball and some form of residue is left behind after the adhesive is removed post-run, does this qualify as damaging the ping pong ball?

Yes, if residue is still on the ball after you remove the adhesive the ball will not be usable by the next team fairly, therefore would be considered damaged. Note, you must be able to clean and return the ball to the event supervisor quickly to allow the next teams to run fairly.