When you play a unit from Kharox Empower, after that played unit dies, does it go to your trash or the enemy's?

It goes to the opponent's trash — you control the unit, but they still own it.

When Kharox's Empower lets you play a unit from your opponent's trash, you gain control of it, but ownership doesn't change. If that unit is killed later, it goes to its owner's trash (Rule 56: a card can never enter a non-board zone belonging to another player, so it returns to the opponent's trash). This applies to any card you play from an opponent's trash, not just Kharox.