No, the oscillation should be on the verge of going unstable.
I doubt it
It depends, is the system on the verge of going unstable?
ZN is awful, it is an attempt to tune a system without understanding it. I posted a link to the Controlguru site that shows the steps on how to tune a heat exchanger using the step method and the techniques to determine the open loop gain, time constant and dead time. This is the way to do it if you don't have auto tuning software.
ZN results in a system that over shoots, under damped. What really gets me is that people try to apply ZN to anything and it just won't work except for simple temperature systems.