In darkest night reads:

"As an additional cost to play In Darkest Night, exhaust four characters you control.

Ongoing: While you control an Emerald Enemies character, characters your opponents control cannot recover during the recovery phase."

So lets say I have 1 Emerald Enemies character. He is stunned along with another non-EE character. If I choose to recover the non-EE character, does my opponent get to recover a character? Or since they all KO at the same time would my opponents not be able to recover a character?

The one free recovery has to happen before stunned characters are KO'd; otherwise, the game wouldn't know which characters were supposed to be KO'd. So you still do control an EE character when the opponent is trying to recover someone, and after his recovery fails all stunned characters will be KO'd.