The rules say that the Dark Side cannot declare engagements during their first turn.

However, may the Dark Side resolve effects that would damage LS objectives?

Say I have 5 (or more) resources. One of my objectives is The Killing Cold .

In my (addmittedly very lucky) first hand I get a Sith Holocron , Executor and Espo Trooper(s) .

I play Sith Holocron for free. Sacrifice it to reduce the cost for Executor to 5, pay the 5 to play Executor then play an Espo Trooper for free.

Can I still in my first turn, sacrifice the Espo Trooper to The Killing Cold and use Executor 's effect to damage one of my opponent's objectives (preferably one of the "as long as this objective is undamaged" ones )?

Sure, there's nothing stopping you.

of course you can damage their objectives on your first turn and their first turn as well. engagements aren't the only way to damage objectives. a dark time for the rebellion is another example of being able to damage your opponents objectives on the first turn of the game. night sister is another.

There are other cards as well. Aggressive Assault is a great first turn play for those pesky 1st turn "while this objective is undamaged" effects.

A Dark Time for the Rebellion is good also. After you refresh, it forces the LS player to damage an objective or a unit. And, they won't have any units to damage yet, and so will have to damage an objective.