Details about driving circuit of mosfet
The resistor is there to control the gate charge current and through that the MOSFET off-to-on time. It's important for the whole bunch of reasons. My personal favorite is that MOSFET going on too fast creates current through the miller capacitance (D to G) and makes itself oscillating (sometimes kills the driver and other circuits, if totally out of control). Other reason is EMI. The resistor of course comes at a tradeoff: the MOSFETs gets hotter.
The diode is there to quickly discharge the MOSFET gate and turn it off.
Pull down resistors ensure that when the driver is off, the MOSFET's gate is discharged. Without it it can pick up noises, randomly open, destroy itself and the other one.