In the past QF have operated a MEL-AKL-LAX and prior to that a BNE-AKL-LAX however these routes have since been dropped. Similarly NZ formerly operated a SYD-LAX service which has also been dropped.
I don't know about any agreement, but I do know Qantas has AKL-CGK.
Qantas do not operate a direct nor nonstop AKL-CGK flight but they do operate SYD-CGK their sole route to Jakarta.
Is there an air rights agreement allowing an Australian carrier to fly internationally from NZ without a continuing segment from Australia?
Straight from the air service agreements between the two governments:
Section 1
ROUTE SCHEDULE
A. Routes for the airline or airlines designated by the Government of New Zealand
1. From points behind New Zealand via New Zealand, and any intermediate point(s), to any point(s) in Australia, and beyond.
2. For all-cargo services, between Australia and any point or points.
B. Routes for the airline or airlines designated by the Government of Australia
1. From points behind Australia via Australia, and any intermediate point(s), to any point(s) in New Zealand, and beyond.
2. For all-cargo services, between New Zealand and any point or points.
C. Routes for SAM Airlines
1. Between points in Australia and/or New Zealand.