If Bringer of Dreams and Nightmares uses Mists of Oblivion, do they get the Power Card's bonus fear for destroying Towns/Cities? (Relatedly, can they trigger Vengeance of the Dead?)

If Bringer of Dreams and Nightmares uses Mists of Oblivion, do they get the Power Card's bonus fear for destroying Towns/Cities? (Relatedly, can they trigger Vengeance of the Dead?)

No. No Towns/Cities are actually destroyed, so Mists of Oblivion grants no bonus Fear. And similarly, Vengeance of the Dead doesn't do any Damage.
To Dream a Thousand Deaths only changes what happens when Bringer of Dreams and Nightmares would destroy/damage Invaders. It doesn't change what happens if an effect checks whether or not Invaders got destroyed.