Thanks for the pics, nothing is jumping out at me I'm afraid. I would disconnect AREF from Vcc, and I would add two 0.1µF caps, one from AREF to ground, and one from AVCC to ground. I'd be surprised if that was causing the problem though.
TeslaIaint:
The motor is just spinning free, and the 7805 is not getting hot. In fact I can't find any part of the circuit getting hot. I tried connecting the 220-ohm resistor and read 4.74V when driven high from output to ground. It measures 0.47V when driven low from output to +5V.
Those voltage readings are good. So if we use a 220-ohm resistor between the MCU and MOSFET gate, and just drive the pin with digitalWrite() rather than PWM, when the pin is driven high, what are the voltages on either side of the 220-ohm resistor?