The international to international minimum connection time in SYD is certainly well less than three hours. As you say, the issue is that the morning flight is NZ-operated.

There is no VA lounge in the SYD international terminal (or any other international terminal). VA partners with NZ for lounge access for business class customers and their own Golds flying trans-Tasman, but DL elites in coach aren’t eligible. You’ll have to enter Australia to access the domestic terminal to use the lounge. With the amount of time you have, if you’re entering Australia, I’d go into Sydney, not visit the lounge.