If "the rest" is zero cards, either because X was 0 or because X was 1 and that card was put onto the battlefield, the library is still shuffled.

Genesis Hydra's first ability will resolve before Genesis Hydra does. Notably, if you put an Aura card onto the battlefield this way, it can't enchant Genesis Hydra.

If you put an Aura onto the battlefield this way, you choose what it enchants as it enters the battlefield. This doesn't target any permanent or player, but it must be able to enchant that permanent or player. For example, you could put a green Aura onto the battlefield enchanting a creature with hexproof controlled by an opponent, but not one with protection from green.

If a card in a player's library has {X} in its mana cost, X is considered to be 0.

A nonland permanent card is an artifact, creature, enchantment, or planeswalker card.

If you have fewer than X cards in your library, you reveal all of them.