Ulamog's first ability resolves independently of Ulamog once they've both been put on the stack. If Ulamog is countered, that triggered ability will still resolve. That triggered ability will always resolve before Ulamog does.

Cards exiled by Ulamog's first and third abilities are exiled face up.

If the player has less than twenty cards in their library, exile all of them. That player won't lose the game until they have to draw a card from an empty library.