Pump control system without arduino
The diode is for reverse voltage protection, and to prevent the 100uF capacitor from discharging back into the motor rather than supplying power to the control circuitry.
The capacitor helps the control circuitry maintain a stable voltage when the motor switches on, which consumes large amounts of current and probably causes the supply voltage to drop.
I'd want to do some detailed analysis to be sure, but it looks like a current limiting resistor for the feedback loop around that comparator.
Because the circuit wouldn't work without them.
Because power MOSFET's consume a lot less power to switch than a relay, and using a relay would have required a MOSFET or BJT anyway, so just use a MOSFET and save the size and money.