Value of resistance used to calculate the MOSFET transductance
This is not calculating the transconductance - this is calculating the voltage gain of the circuit.

gm is the transconductance of the FET and is the change in current through the device caused by a unit change of input voltage.

The output voltage is the gm multiplied by the effective output resistance.

The effective output resistance consists of the resistor Rd in parallel with the internal resistance of the FET (this is omitted in your calculation).  This is normally referred to as Ro.