The regeneration shield will last until the turn ends or until it’s used.

You can activate Gore Vassal’s ability targeting itself, but it won’t get a -1/-1 counter or regenerate because it’ll be an illegal target when the ability resolves.

Let’s say you activate Gore Vassal’s ability targeting a creature with 3 toughness and 2 damage marked on it. The regeneration shield will be created just before the creature is destroyed for having damage marked on it equal to or greater than its toughness. The creature will then regenerate. That is, all damage will be removed from it and it will become tapped; any counters on the creature, including the -1/-1 counter put on by this ability, remain.