If player one area effects (1) every opposing unit and a unit dies then player 2 interrrupts and puts down one or more Elysian Assault Teams, would they also die?

No.

Remember that damage us dealt in a 3-step process:

1. Assign damage

2. Shield

3. Place damage

When a unit uses its Area Effect, all the opponent's units go through these things together. So, all units have damage assigned simultaneously in #1. Then shields are played (1 at a time, 1 for each unit) in #2. Then, all unshielded damage is placed on the corresponding unit simultaneously in #3.

You interrupt #3 - where the damage is placed and the unit potentially destroyed - with Elysian Assault Team. At that point, it is too late for the Area Effect to go back to #1 and assign more damage to the unit that just entered play.

Effectively, since the Elysian Assault Team is not in play when the Area Effect assigns damage, Area Effect misses the opportunity to put any damage on it.