Do I need a gate resistor and/or a pull-down resistor for a TC4420 (or any gate driver) driving a MOSFET?
The gate driver actively pulls the gate voltage high or low, so a pull-down resistor is not needed.
The gate driver datasheet says that, due to its low output resistance, it may be a good idea to add some sort of short-circuit protection, e.g. the MOSFET could fail as a short, so I suggest that you calculate the gate resistor value to limit the power dissipation in the driver to whatever your design can cope with.
Further reading:

Do MOSFETs usually burn open or closed?
Calculating the current needed to drive an N-MOSFET