What are the implications and dangers of a 'stuck flap' that necessitates a diversion to closest airport?
In the most likely scenario, the flaps are not physically stuck, but rather have been deactivated by the detection of some degree of asymmetry. It can happen on either extension or retraction.
Your maximum speed is very limited. If the flaps tripped when selected from 20 to 30, you'd be limited to the flap 30 extension limit.
Drag is increased.
Maximum altitude is FL200.
FMC fuel predictions are unreliable.
Approach speed will be higher than normal. The one time it happened to me (in a 767), the increase was 40 knots.
You will need the longest runway you can find (or the biggest headwind). Both would be good. Palmerston is almost 2,000' shorter than Ohakea.
Diversion fuel is not based upon odd configurations. If the aircraft had diversion fuel for Auckland, it would not have been enough in this configuration.
The ability of the airport to handle the passengers, or even the aircraft itself, is not a consideration.