Before/after the dates mentioned above, or in between?

If you're looking at in between I wouldnt bother. You only have 10 or 11 days in Australia and in that period travelling the equivalent of Washington to LA. There's plenty to see and do in Australia - will be rushed to try and fit a trip to NZ in.

If before/after, and have the time, then may as well - it is a long way to come to NZ from USA but only a few hours from Australia.

Winter sports will be available then, especially in the south island and central north island. Weather is cool and wet - so factor that in. But doesnt get too cold - snow is only in the mountains for example.