If ackbar is put into play after an engagement has begun, but before the edge battle, and does 1 damage to all opposing participating units - destroying them- Does the edge battle still occur?
With trench run in play, if the death star was attacked by Home One, does Home One still do 1 damage to all other objectives?
1. Yes, however only players with units participating in the engagement can place cards in the edge battle, so if Ackbar eliminated all of your opponent's participating units only you will be able to place cards.
2. Yes. Home One places damage on all objectives that are not currently engaged when it strikes as an attacker. Nothing there requires it to be attacking an objective.