Do leader abilities apply for the entire encounter?

Yes, leader abilities do continue working despite the death of the leader.

I have just verified this (at least for the case of an opponent's leader dying) by facing a player in the Arena with Captain Phasma as their leader, specifically targeting and taking out Captain Phasma, then noting that on at least 3 occasions during the rest of the fight other characters were given bonus attacks (characters that do not usually have a way of receiving bonus attacks such as Luminara Unduli and Darth Sidious).

In the case of Galactic War, if you kill the leader but fail to kill the other enemies before your team is killed, when you go in with your next team the leader ability will no longer apply.