If you have more than one unit engaged to an objective and one unit kills it, it's immediately taken out right? So then would the attackers other units in the engagement not be able to resolve their icons because there's no objective anymore?

Their icons may not do anything, although tactics would still work fine, but every unit involved must still be exhausted to strike anyway. Any blast damage would go nowhere.

So what about even unit damage against defenders? Can you still resolve them even tho you just destroyed the objective?

Yes. Every unit must strike, every icon resolves, regardless of whether the objective is destroyed or not. After the objective is destroyed, even blast damage icons still resolve ... it's just that their resolution no longer applies damage because the objective's no longer there.

If a card says after an objective is destroyed can that be after the engagement as a whole or must it be immediately after the objective leaves play?

If a card says after an objective is destroyed can that be after the engagement as a whole or must it be immediately after the objective leaves play?

Immediately

Every engaged unit must strike. No unit can leave the engagement without completing its strike(s), given that it is not already focused and therefore unable to strike.

Every engaged unit must strike. No unit can leave the engagement without completing its strike(s).

Not quite true. Every engaged unit must have at least one focus token on it before you can leave the resolves strike step of the engagement. Whether it gets the focus token(s) from striking or some other effect (often opponent's Tactics icons) does not matter. You loop through the resolve strikes flowchart until all engaged units are exhausted.

Every engaged unit must strike. No unit can leave the engagement without completing its strike(s).

Not quite true. Every engaged unit must have at least one focus token on it before you can leave the resolves strike step of the engagement. Whether it gets the focus token(s) from striking or some other effect (often opponent's Tactics icons) does not matter. You loop through the resolve strikes flowchart until all engaged units are exhausted.

Of course. I edited my above post.

Sometimes I see the question asked by new players as to whether you can choose not to strike, etc. Given that a unit is not already focused, it has to strike before it can leave the engagement.

Understood. I just wanted to bring it up because I've also seen the misunderstanding that every unit must strike even if it is already focused. :-)

This means that Red 2 focus to strike, and destroys the objective, it must focus to strike again, right?

Yes.

One trick for Red Two is planning for an unopposed bonus to finish off the objective, because that happens after all strikes so Red Two is ready after the engagement.