If you cast a creature with bloodthirst while Leyline of Lightning is on the battlefield, you may use the Leyline's ability to deal damage to an opponent. If you do, that damage is dealt before the creature spell resolves, and the creature will enter with counters from its bloodthirst ability.

A player's "opening hand" is the hand of cards the player has after all players have taken mulligans. If players have any cards in hand that allow actions to be taken with them from a player's opening hand, the starting player takes all such actions first in any order, followed by each other player in turn order. Then the first turn begins.

While resolving Leyline of Lightning's second ability, you can't pay multiple times to deal damage multiple times.