You choose whether to cast the target card while Emet-Selch of the Third Seat's second ability is still on the stack. You can't wait to cast it later in the turn. Timing restrictions based on the card's type are ignored.

The cost reduction applies only to generic mana in the total cost of spells you cast from your graveyard.

To determine the total cost of a spell, start with the mana cost or alternative cost you're paying (such as a flashback cost), add any cost increases (such as kicker costs), then apply any cost reductions (such as that of Emet-Selch of the Third Seat's first ability). The mana value of the spell is determined by only its mana cost, no matter what the total cost to cast that spell was.