A player who has a card in hand can't choose to lose 3 life instead of discarding a card.

The value of X is determined only as Liliana's second ability resolves. Once that happens, the value of X won't change later in the turn even if the number of cards in your graveyard changes.

As Liliana's first ability resolves, first the player whose turn it is chooses a card in hand without revealing it, then each other player in turn order does the same. Then all the chosen cards are discarded at the same time. Finally, each opponent who couldn't discard a card loses 3 life.

If you have no cards in hand, you just don't discard anything. You don't lose 3 life. Other players still discard if able.

In a multiplayer game, if a player leaves the game, all cards that player owns leave as well. If you leave the game, any permanents you control from Liliana's emblem are exiled.

Creatures put onto the battlefield with Liliana's emblem gain haste indefinitely.