Emrakul, the World Anew

Madness works independently of why you're discarding the card. You could discard it to pay a cost, because a spell or ability tells you to, or because you have too many cards in your hand during your cleanup step. You can't discard a card with madness just because you want to, though.

If you discard a card with madness, you discard it into exile instead of into your graveyard. When you do, you can either cast it from exile for its madness cost or put it into your graveyard.

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.

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.

Casting a spell with madness ignores the timing rules based on the card's card type. For example, you can cast a creature with madness if you discard it during an opponent's turn.

If you discard a card with madness while a spell or ability is resolving, it moves immediately to exile. Continue resolving that spell or ability, noting that the card you discarded is not in your graveyard at this time. Its madness triggered ability will be placed onto the stack once that spell or ability has completely resolved.