He's being extradited now, which means that it's US laws that apply now. I don't think this particular fact is evidence of any law breaking, unless the email the user sent said they were pirating the show. Because it's hard to prove that Kim knew what "Dexter" was, and even if he did, it's hard to prove he didn't assume it was uploaded legitimately (by Showtime, for example).

That said, there's a little more to it than just promptly addressing DMCA takedown requests. In order to be eligible for the DMCA safe-harbor, a company must not be aware of specific cases of copyright infringement on their website without taking some action agais them.

Technically he has the right to appeal, which I'm sure he will do, so he's not being extradited just yet and New Zealand laws still apply.

"Proving" something in a court of law (in criminal law) only requires that there is no reasonable doubt. Providing alternative explanations that are not logically impossible, but exceedingly unlikely, does not help you much. If the Dexter thing occured as written here, proving intentional copyright infringement is very easy.

Assuming they are allowed to use the evidence with how it apparently was gathered. And then there are much bigger problems in the society - however the US got away with torture camps/prisons and other scariness, so "surely" they'll get away with this unless the New Zealanders do something about it.