Can Bringer of Dreams and Nightmares destroy already-damaged pieces by moving them into/out of lands that affect their Health? (Eg: 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 moving them into/out of lands that affect their Health? (Eg: 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 damage or destroy effects, which is what To Dream a Thousand Deaths modifies; it's just changing where a piece is. 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 will trigger after the action (Power) is complete.