With Thornbite Staff, do you need to pay two mana to untap this card if a creature is put into the grave yard or is the mana cost only for doing one damage to target creature or player?
The mana cost is only for the first ability, dealing the damage.
Thornbite Staff gives the creature 2 complete separate and unrelated abilities. The first is an activated ability; because it has the format {cost}:{effect}. The second is a triggered abilitty, because it starts with the word “whenever”. Each ability is surrounded separately by quotes, with “and” in between.