Rulings

You may ignore a Trap's alternative cost condition and simply cast it for its normal mana cost. This is true even if its alternative cost condition has been met.
Casting a Trap by paying its alternative cost doesn't change its mana cost or mana value. The only difference is the cost you actually pay.
Effects that increase or reduce the cost to cast a Trap will apply to whichever cost you chose to pay.
Mindbreak Trap's alternative cost condition checks whether an opponent cast three or more spells this turn, not whether those spells have resolved.
If a spell is exiled, it's removed from the stack and thus will not resolve. The spell isn't countered; it just no longer exists. This works on spells that can't be countered, such as Terra Stomper.