For Mindbreak Trap's alternative cost to apply, a single opponent must cast three or more spells. Three opponents each casting a single spell won't work, for example.

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.