If I use Cull of the Weak and my opponent plays Thrill of the Hunt on the unit he chose, does my opponent get to not kill any units?

No, your opponent does not get to skip killing a unit.

Here's the sequence: Cull the Weak doesn't target, so no unit is chosen when it's played. Your opponent can respond with Thrill of the Hunt — that banishes their unit and plays it back to a battlefield. But when Cull the Weak resolves, they must still "kill one of their units." If that replayed unit is their only unit on board, they are required to choose that same unit to kill. They cannot opt out as long as any unit of theirs exists. (FAQ 10077, 11012)

The only way they'd skip it is if they somehow had zero units on board at resolution — but Thrill of the Hunt puts the unit back on board, so that generally won't happen.