ESD tests: when device is powered and not
Some answers:


When ESD testing ICs they are powered OFF. For an IC most ESD problems can occur when it's not yet soldered on a PCB so during handling. It would make the ESD test more complex and not have any advantages to ESD test with the IC powered up. An ESD pulse is such a stressfull event that it doesn't matter much if the IC is on or not. What matters is that the ESD protection handles the pulse and dissipates the energy in a non-damaging way.
A finished product is usually tested both on and off. When on it can be checked that the product remains functional. Maybe it can enter a locked-up state or reset which might be acceptable or not.
I don't see how an ESD test on a powered device makes much difference in relation to damage to the device. The ESD pulse is so destructive when not dissipated properly it will do damage anyway.
Yes on a PCB the decoupling should absorb the ESD pulse, if you need extra protection (there's not much decoupling for example) you might want to add some device to absorb the pulse on the power rails. I don't think a zener diode is fast enough, there are certainly devices available that will be fast enough though. On all ICs there are so called crow-bar circuits which short the power supply lines when the voltage rises too quickly (as in the case of an ESD pulse). Also there are diodes that protect against the supply from being reversed polarity. An ESD pulse can be positive or negative !
The charge of an ESD pulse is not very large so on an board be it powered or not, after the ESD pulse is absorbed you will NOT notice any voltage change on the power rails. So do not worry about the ESD pulse accumulating on the power lines, the charge is simply too small to notice.