You may exert Clockwork Droid only as you attack with it. You can't do so later in combat, and creatures put onto the battlefield attacking can't be exerted. The ability that triggers on exerting it will resolve before blockers are declared.

If an exerted creature is already untapped during your next untap step (most likely because it had vigilance or an effect untapped it), exert's effect preventing it from untapping expires without having done anything.

You can't exert a creature unless an effect allows you to do so. Similar effects that "tap and freeze" a creature don't exert that creature.

If you gain control of another player's creature until end of turn and exert it, and then that player regains control of it, it will untap during that player's untap step.