If I trigger Dazzling Aurora and draw Rengar Trophy Hunter and my opponent has a battlefield occupied, can I play him to their battlefield and choose who I deal damage to?

Yes, you can play Rengar, Trophy Hunter to an enemy-occupied battlefield off Dazzling Aurora's trigger — the FAQ directly confirms this (RiftJudge FAQ #11156). Rengar's unique permission ("I can Ambush to a battlefield where there are enemy units, even if you don't have units there") overrides the normal restriction that would otherwise prevent you from playing a unit to a battlefield you don't control.

However, Rengar doesn't have an ability that deals damage when played or Ambushed. His card only grants permission to enter battlefields with enemy units — there's no "deal damage to a target" on his text. So there's no damage to choose or assign when he enters play via Aurora. Once he's there, he'd be the Defender at that contested battlefield, and any damage would occur during the subsequent combat step like normal.