Can I play Rengar Trophy Hunter as a reaction to a temporary unit dying on upkeep to conquer on my opponent's turn? If so, does a showdown start before the unit dies or can he enter the battlefield empty?

Yes — but only because you react while the unit is still alive. Rengar never enters an empty battlefield; he enters first, then the unit dies.

The Temporary trigger (Rule 816: "At start of controller's Beginning Phase... kill this") goes on the Chain during your opponent's Beginning Phase, putting the game in a Closed State — your window to play Rengar as a [Reaction]. Since your play resolves after the trigger is on the Chain, it resolves first (LIFO), so Rengar enters while the enemy unit is still present, satisfying his Ambush condition. Only then does the kill trigger resolve and remove the unit.

No showdown starts before the unit dies. The showdown is staged in the Cleanup that follows the chain emptying (Rule 323.8) — by then only Rengar remains, so it's a non-combat showdown, you establish control, and you Conquer (Rule 348.2.a.1) as long as you haven't already scored that battlefield this turn. Conquering on your opponent's turn is fine. (If you waited until after the unit died, the battlefield would be open and Rengar couldn't be played there at all.)