What is the purpose of a capacitor placed between the Base and Emitter of NPN Transistor & Zener Diode placed between Gate Source of MOSFET
Adding a capacitor between the gate and emitter will influence the switching behavior of the IGBT. Its job is to take up additional charge originating from the Miller capacitance. This increases the required driver power and the IGBT exhibits higher switching losses for the same gate resistor.

By increasing the gate-emitter capacitance directly at the IGBT, the peak fault current is reduced. In other words, the transistor can not turn on by iteslf.
https://www.controleng.com/articles/tutorial-mitigating-parasitic-turn-on-effect-in-igbt-output-drives-to-improve-drive-performance/
https://www.researchgate.net/figure/By-increasing-the-gate-emitter-capacitance-directly-at-the-IGBT-the-peak-fault-current_fig6_228894736

The zener diode sustains the voltage on the gate stable(approximately), which is necessary to control the MOS. Its a standard circuit, it has the same purpose as a resistor divider, except that the voltage through the zener will always be the same, even when the voltage of the source changes. As long as the voltage of the source does not fall to much, in which case the voltage on the gate will be the same as the source and the zener will not turn on.