Dusknoir

If Dusknoir uses its "Cursed Blast" Ability and targets an opponent's Benched Pokemon while Battle Cage is in play, would Dusknoir be Knocked Out even though it placed no damage counters?

Yes. It still used the Ability even though the placement of the damage counters was blocked by Battle Cage. So Dusknoir is Knocked Out.

If both players have only one Prize card left and my opponent's Dusknoir uses its "Cursed Blast" Ability to Knock Out my Pokemon, who wins?

If one player has a Benched Pokemon to put into the Active Spot and the other doesn't, the player who can send out a new Pokemon wins. If both players can or neither player can the game goes to a Tiebreaker Game, or if there is no time left the game counts as incomplete.

If Dusknoir is Active and uses its "Cursed Blast" Ability to put damage counters on the opponent's Active Pokemon resulting in a Knock Out, which player puts out a new Active Pokemon first?

The player whose turn would be next takes their prizes first and then the other player takes their prizes. Then the player whose turn would be next promotes a new Active Pokemon if needed, followed by the other player if they need to promote a new Active Pokemon.

If Dusknoir is Active and uses its "Cursed Blast" Ability to KO a Togekiss with "Wonder Kiss" Ability, would Togekiss still get to flip for Dusknoir being KO'd or not?

Yes, you would get to flip for Wonder Kiss. All effects have to be resolved before resolving KO's.

If Dusknoir or Dusclops is in the Active Spot and uses its "Cursed Blast" Ability to KO a Togekiss with "Wonder Kiss" Ability, does Togekiss get to take an extra Prize Card or not?

If the Dusknoir or Dusclops is active when they use Cursed Blast, Wonder Kiss will apply.