Do N-channel MOSFETs require a pull-down resistor?
Yes, if you just tristate after pulling it high, then the gate will stay floating high. You either need a resistor to pull it down to ground or you need the input signal to drive it low.

The resistor can/should be relatively high valued compared to your input resistor to prevent excessive voltage drop when you have it set as a high input. You only have to drain the inherent capacitance on the MOSFET gate when you're pulling it low so even at a high resistance to ground the RC time constant is usually relatively short.