The “if you do” part is only done if you successfully sacrifice this card. In other words, it has to be on the battlefield when the trigger resolves or it does not deal damage.

The choice of which card to name is part of the resolution of the spell or ability which puts Booby Trap onto the battlefield. It can’t be countered.

If an effect causes the player to draw multiple cards, each card is revealed as it’s drawn. If Booby Trap’s ability triggers, it will go on the stack once the card-drawing effect has concluded.

If Booby Trap’s last ability is countered, Booby Trap remains on the battlefield (and no damage is dealt).

If a single Booby Trap’s ability triggers multiple times, then only the first ability to resolve causes Booby Trap to be sacrificed and damage to be dealt.