I play LS and engage a DS objective with only one unit. DS player declares his only non-focused unit (with 1 HP) to defend.

In the action window before the Edge Battle, I played Admiral Ackbar from my hand, doing 1 damage to the DS defending unit and destroying it.

The question is: Now that the defending player has no more units before the Edge Battle begins, can they still participate in the edge battle?

Core Rulebook, pg 18:

"A player must control at least one participating unit to place cards in the edge battle."

FAQ 1.3, pg 4

"If they [units in an engagement] are removed from an engagement, there is no memory of that unit having participated in that engagement."

If I read the above rules correctly, if all declared attackers/defenders on one side are destroyed in the action window before the Edge Battle begins, that side has no participating units and can't place cards in the Edge Battle.

Is this reading of the rules correct?

You are correct.

I have a similar question. I play LS and the DS player attacks me with a unit with only 1 HP. I play Admiral Ackbar as my only defender and destroy it. Is there still an edge battle and combat to follow so that I can use his two combat icons to focus his other units?

And If there is an edge battle, he cannot play any cards correct?

As above,

If you play Ackbar immediately after he declares the attacking unit.

"A player must control at least one participating unit to place cards in the edge battle."

If you don't there is an edge battle and you can then play Ackbar in the next action window.

But if I wait to play Ackbar, then I will loose the edge battle correct? Because if I do not have any defenders then I cannot play any cards in it.

Basically, I want to play Ackbar to kill his guy, then use Ackbar's two tactics combat icons (which are edge dependant, so I need to win the edge battle) to focus two of his other units. But he contends that if I kill all of his attackers before the edge battle, then the combat stops and I do not get to use his combat icons because there is never a fight.

Killing all participating units on either side does not end an engagement. Even if all participating units on both sides are eliminated the engagement continues as normal (in that case, the attacker wins the edge battle as there are no defenders... Relevant if Hit and Run is in play).

So if I play Ackbar before the edge battle, it will kill his attacking unit. Then I will win the edge battle (because the DS player doesn't have any units participating) even though I am the defender, and get to use Ackbar's edge dependent combat icons correct?

Yes as attacker or defender (attacker wins automatically when there are no defenders) and the defender wins on a tie.