Can someone explain this high-speed P-channel MOSFET drive circuitry?
The zener is there for to stop the negative pulse through Ch from exceeding the maximum gate to source voltage, on positive pulses it is simply a diode.
The circuit load is capacitive coupled therefore it will only energise the load while there are pulses.  Both Inh and Inl require pulses not a constant DC voltage, also Inl must be low if sending a pulse to Inh, hence if pulsing both pins they should be 180 degrees out of phase with each other, otherwise you will destroy the ouput stages.  Check the device datasheet to confirm it's operation as I am asuming that the gate drive IC is to enable pulses from a microcontroller.  If this is a single device then you should have included the part number in your description.