If it were a single Power doing 2 Damage then 1 Damage, it would earn the Fear, because then that Power would have destroyed an unhurt City. For each Power Bringer uses, briefly imagine a universe where you resolve the entire Power normally. Whenever it would destroy an Invader in that universe, in the real game you earn 0/2/5 Fear and Push it if it's an Explorer or Town. Tags: Damage, Bringer of Dreams and Nightmares, To Dream a Thousand Deaths (Bringer of Dreams and Nightmares)
This also means Bringer's Repeat cannot usefully combine Damage with its original – for instance a Repeated "2 Damage" Power couldn't do anything to an undamaged City. The original action would resolve, and 2 Damage isn't enough to destroy the City, so nothing happens. Then the second use of the Power resolves, and 2 Damage isn't enough to destroy the City, so nothing happens. (Discussion elsewhere) Tags: Repeat, To Dream a Thousand Deaths (Bringer of Dreams and Nightmares), Bringer of Dreams and Nightmares
This isn't necessarily obvious: To Dream a Thousand Deaths affects the instructions of Bringer's powers (see FAQ), and unlike most damage modifiers, Badlands applies when the results of an action deal Damage (otherwise "Badlands don't apply if Damage is reduced to 0" makes no sense). So on first blush, Badlands shouldn't apply.
Can Bringer of Dreams and Nightmares combine Damage-dealing Powers usefully with Damage from other Spirits?
It depends on ordering. Bringer of Dreams and Nightmares can take advantage of partially Damaged Invaders: e.g.: if another Spirit does 2 Damage to a City, then Bringer of Dreams and Nightmares could afterwards use a 1-Damage Power to earn 5 Fear (because 1 Damage would have destroyed the City). It could even use several Damaging Powers in a row to earn 5 Fear for each! But if Bringer uses a 1-Damage Power on an unhurt City, it will do nothing, and have no lingering effects that would interact with anything done later.
Can Bringer of Dreams and Nightmares do real damage (destroying cities, etc) with Power X?
The answer, in general, is 'no'. If the Power says "damage" or "destroy", it doesn't matter if it also says that Dahan, Beasts, Invaders, or anything else are doing that damage, the source is still ultimately Bringer of Dreams and Nightmares and is only nightmare damage.
This applies even if the power has some other spirit doing something (eg, Blazing Renewal). If Bringer of Dreams and Nightmares played the power, it cannot do real damage.
Thematically, the Dahan are actually doing a raid in the real world, it is not just a nightmare caused by Bringer. It also doesn't matter if it's a choice Event you pay Energy for: the damage is still real.