How does the secret objective requiring you to have a ship in every wormhole system and the Ghosts of Creuss react? Do I have to capture both their home systems? Must I also control the system in which they create their wormhole using technology?

Also do you play Moment of Triumph during the space battle or can you play it after the space battle is over? If it is played after the battle how does this interact with Communication Breakdown which forbids the usage of action cards during a space battle?

How does the secret objective requiring you to have a ship in every wormhole system and the Ghosts of Creuss react? Do I have to capture both their home systems? Must I also control the system in which they create their wormhole using technology?

IIRC, the Creuss wormholes are not included in any wormhole-related SOs. I think this one is in the official FAQ, if you want to confirm that.

Also do you play Moment of Triumph during the space battle or can you play it after the space battle is over? If it is played after the battle how does this interact with Communication Breakdown which forbids the usage of action cards during a space battle?

I'm not familiar with the text from either of these cards, but "after the battle" is not "during the battle," generally speaking.

As Steve-O said, the "D" wormholes are not required for the wormhole-controlling SOs.

I think Moment of Triumph is required to be played "immediately after" destroying an opponent's flagship or war sun, right? That would be during the battle, then, and so couldn't be played if Communications Breakdown is in effect.

Moment of Triumph is played "after an opponents flagship or war sun was destroyed", we rules that since the ship was destroyed during combat the card would need to be played during the battle and thus was blocked by communication breakdown.

I agree with the above on Moment of Triumph - you have to play it after the ship is destroyed, which means RIGHT after. If Communication Breakdown is in effect, you can't play it.

Good, that's how we ruled in the game. Much to the players anguish!