Two questions:

Batman Earth 2 - Exhaust an Earth 2 character you control, discard a card -> Target player chooses a non-Team-Up location and non-Team-Up ongoing plot twist he controls, then replaces one of those cards. Use only once per turn.

This only works if they have a non-Team-Up location and non-Team-Up ongoing plot twist? If they only have 1 flipped it does nothing?

also:

Mourn for the Lost - To play, discard a character card. Negate target non-triggered effect from a plot twist with cost 1 or less. If you control a Gotham Knights character, draw a card.

I don't understand what a non-triggered effect is? Like a plot twist without an additional payment (like exhaust someone or stun someone or something?)

Batman's change in wording was specifically to ensure it would work even if the opponent didn't control one of each. Remember, you must resolve as much of an effect as possible---even if you don't control a location, you still have to pick an ongoing plot twist, and vice versa.

As for Mourn for the Lost, triggered is a type of power, marked by having one of the phrases "When," "Whenever," or "At the start of."