Q1, D1, R2 is an active pull-up circuit, designed to turn the MOSFET off faster than the other MOSFET turns on. When Q4 sinks current, it turns on the p-chan, upper MOSFET by sinking current through the diode. When Q4 does not sink current, Q1 actively pulls the gate voltage high, turning Q2 off quickly. During a transition, the capacitance of the Q2 gate, is charged by Q1, or discharged by Q4.R16 and R20 set the gain of the amplifier to about 10, by attenuating the feedback signal. C13 is not really a low-pass, it is a high-pass that speeds up the feedback for high frequencies, probably to improve stability.Current through R19 is set by Q4 base and emitter voltage, that current flows through R2, providing or dropping sufficient voltage to turn Q2 on. The input voltage only needs to be about 8 volts, since that will sink enough current through R19 to provide enough gate voltage across R2 to turn on the P-chan Q2.