You need a pull-up resistor on the reset pin for the same reason you need to pull down resistors on the gate of the MOSFET.
Yes and no. A pull-up resistor is required - but not always an
external pull-up resistor. Many microcontroller have them internally already. For instance most modern PIC's and almost all AVR's have them.
Have a look at the datasheet for ATmega48A, ATmega48PA, ATmega88A, ATmega88PA, ATmega168A, ATmega1688PA, ATmega328, ATmega328P where you clearly can see that there is an internal pullup already at the reset pin.
According to the graphs later on in the doc it will pull about 100uA when vcc i s 5 volts which would give it a ballpark value of 50k.