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 deal 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 actually got destroyed.