It still deals damage to an additional unit, hero, or building. This damage happens at the same time as the damage it deals to the defender (and usually at the same time as the damage the defender deals to the unit with overpower).
This is correct.
For the Swift Strike thing, if you attack my Glorious Ninja with 2 damage on it using your Centaur, your Centaur will not deal damage to anything. My Glorious Ninja will survive the attack.