If the LED on the RAMPS board lights up, the mosfet is at least passing minimal current. Take the mosfet out of the equation, just hold the ground from the hot end to the power supply ground. If it heats up, then it's either D10's connector, the mosfet, or a trace/solder joint. A continuty test and reflowing the joints should eliminate all but the Mosfet as the issue.
I'd also check your hot end to make sure that it doesn't have a short in the heater/resistor. Measure the resistance between the positive lead at the step down and ground. It should measure a few ohms (depending on what you use as the heating element) When D10 turns on (grounds), it may allow a direct short that trips your step down (why the fan shuts off) and puts a different load on your power supply (why the PSU fan changes speed). When it momentarily worked again, that short to ground might have been broken allowing it to work again...for a short period).
I'd also check your hot end to make sure that it doesn't have a short in the heater/resistor. Measure the resistance between the positive lead at the step down and ground. It should measure a few ohms (depending on what you use as the heating element) When D10 turns on (grounds), it may allow a direct short that trips your step down (why the fan shuts off) and puts a different load on your power supply (why the PSU fan changes speed). When it momentarily worked again, that short to ground might have been broken allowing it to work again...for a short period).