The pull-down resistor is only to set the default state while the MCU is waking up (most go tri-state after reset). It can (indeed, should, for this reason) be placed before the series resistor, thus losing no voltage at the gate.
The series resistor serves to slow the gate risetime, against gate capacitance; this can be harmful as it increases switching losses, but also helpful as it slows switching in general, so can avoid problems like generating radio interference or needing protective circuitry to address transient currents or voltages. The value or range required, depends on the application.
Note that the MCU's pins have an approx. 20-50 ohm source resistance themselves. This isn't documented directly, but can be inferred from the VOH and VOL parameters, and what currents they are tested at. This resistance limits current some already. When driving small capacitors (other CMOS pins, trace impedance, and small MOSFETs too), this resistance alone is sufficient -- the current is not continuous so it doesn't heat up the device or cause damage. An external gate resistor is still recommended, particularly if there is a long distance between MCU and MOSFET, as the MOSFET can oscillate as it switches (another source of radio interference), and a resistor placed at the MOSFET gate prevents oscillation.
There are less common reasons you might want a gate resistor, too. For example, if you want to prevent damage to the MCU in event the MOSFET fails catastrophically. (Typically when a transistor fails, it becomes a three-way short, at least momentarily; drain supply voltage can thus flow into the gate circuit.) A series resistor, with a clamping TVS from source to gate, can be used to shunt this energy to ground, away from the MCU. (This isn't very important most times, as any electrical failure means board replacement, and survival of the MCU is irrelevant; but it can be handy while prototyping, when transistor failures are likely.)