Emrakul, the World Anew

The control-change effect of Emrakul's first ability lasts indefinitely. It doesn't wear off during the cleanup step or when Emrakul leaves the battlefield.

If you discard a card with madness to pay the cost of a spell or activated ability, that card's madness triggered ability (and the spell that card becomes, if you choose to cast it) will resolve before the spell or ability the discard paid for.

A card with madness that's discarded counts as having been discarded even though it's put into exile rather than a graveyard. If it was discarded to pay a cost, that cost is still paid. Abilities that trigger when a card is discarded will still trigger.