According to the Law of Conservation of Momentum, the total momentum must be the same before and after a collision. The total momentum before the bat hit the ball was 12 kg m/s. After the collision, the ball has a momentum of 8 kg m/s. Therefore, the bat must have a momentum of 4 kg m/s.
Let's use Newton's Law of Second Motion: F=ma. When no other direct force is acting on the system, the acceleration is due to the gravity. The modified equation becomes: F = mg. So, yes, you need to take into account the gravitational accelerations in the moon and on Earth.