If an Opponent has a Trample monster on their side of the field, and I equip them with My own Fairy Meteor Crush or Big bang Shot, Who loses the Life Points? I'm assuming that both players do. With the monsters own effect taking Life Points from me, and the Equip Card allowing me to take Life Points from the opponent. Any other Interpretations?

Only your opponent would take the damage.

Fairy Meteor Crush
When a monster equipped with this card attacks with an ATK that is higher than the DEF of a Defense Position monster, inflict the difference as Battle Damage to your opponent's Life Points.

Rulings
If you equip your "Fairy Meteor Crush" to an opponent's monster, and it attacks a Defense Position monster on your side of the field with ATK greater than your monster's DEF, the opponent will take the difference as Battle Damage. This is because "your opponent" is the opponent of the player who controls "Fairy Meteor Crush" not necessarily the opponent of the player who controls the monster.

Your opponent will receive the damage, as stated in the cards.

He already acknowledged being aware of that ruling, however, what then causes the opponent's monster's trample effect to be cancelled?

both players will take damage. just because fairy meteor crush is the more recent effect it does nothing to interfere with the monsters own trample effect.

both effects will work normally.

timestamping is done in situations like level limit/final attack orders situation where both cannot possibly function at the same time. this isn't a situation like that. no part of either effect prevents the other from occuring.