Can Bringer of Dreams and Nightmares destroy already-damaged pieces by giving them a Health penalty or causing them to lose a Health bonus? (E,g,. moving a Dahan that has taken 2+ Damage out of a land where it's getting a Health bonus from "Promises of Protection.")

Can Bringer of Dreams and Nightmares destroy already-damaged pieces by giving them a Health penalty or causing them to lose a Health bonus? (E,g,. moving a Dahan that has taken 2+ Damage out of a land where it's getting a Health bonus from "Promises of Protection.")

Yes. The Power itself isn't performing any explicit "damage" or "destroy" effects, which is what "To Dream a Thousand Deaths" modifies; it's just changing the Health of the piece. From that, the game's core rules cause the piece to be destroyed.
The destruction does still happen within the scope of the action (Power), so triggered actions like "Vengeance of the Dead" trigger after the action (Power) is complete.
The most common ways for this to happen are: