If my opponent moves to an empty battlefield on their turn, can I play the new orange Rengar to that battlefield? And am I the defender?

Yes, you can play Rengar, Trophy Hunter to that battlefield, and you will be designated as the Defender.

According to RiftJudge FAQ #9820, here is the breakdown of why this is the case:

1. Playing Rengar, Trophy Hunter

When your opponent moves a unit to an open (unoccupied and uncontrolled) battlefield, they initiate a non-combat showdown and that battlefield becomes Contested. Because there is now an enemy unit present at that location, you satisfy the requirements of Rengar, Trophy Hunter’s ability (which allows him to be played to a battlefield with enemy units, even if you have no units there). Therefore, you may play him there as a [Reaction] during the ongoing showdown.

2. Attacker and Defender Designations

• Attacker: Your opponent is the Attacker because they were the player who first moved their unit to the open battlefield, causing it to become Contested (Rule 442.1.a.1).
• Defender: By playing Rengar, Trophy Hunter to that same contested battlefield while your opponent's unit is present, you become the Defender (Rule 442.1.a.2).

Summary of the Sequence

1. Opponent's Move: Your opponent moves a unit to an open battlefield. It becomes Contested, initiating a non-combat showdown. Your opponent is the Attacker.
2. Your Response: During this active showdown, you play Rengar, Trophy Hunter as a [Reaction]. Because there is an enemy unit at the battlefield, you may play him there despite having no other units present.
3. Designation: Upon playing Rengar to the contested battlefield, you are designated as the Defender.
4. Showdown Continues: The showdown proceeds. If units from both players remain, a combat will begin once the chain clears and the game returns to a neutral open state.

This interpretation is based on the rules for Contested status (Rule 445/446), the definitions of Attacker/Defender designations (Rule 442.1.a), and the specific conditions for Rengar, Trophy Hunter provided in RiftJudge FAQ #9820.

Note: I could not fully verify this ruling against the retrieved rules, as the interaction with the Ambush keyword remains ambiguous.