This effect does not specify "and sends it to the GY". It triggers at the end of the Damage Step, even if destroyed after Damage Calculation (either by battle or by effect), as long as it destroyed a monster by battle.
This effect activates at the end of the Damage Step.
The second effect targets 1 Set card on the field.
Even if you activate the first effect of this card, you do not have to attack again in a row.
If both this and the opponent's monster are destroyed by battle with each other, you can still activate the second or third effect of this card.