Can I drive all these FETs from this microcontroller?
I wouldn't trust that the microcontroller distributes the current for you. If you draw to much current you'll exceed the maximum power rating, and this can blow the chip.

However, how about adding a series resistor between each GPIO pin and each FET gate? 

You have a budget of about 1.78mA per pin (150 mA /84 pins). With a supply voltage of 5V that'll give you a resistance of 2.8 kOhm. 3.1k would be the next higher standard value. You want to be on the safe side, right?

For 3.3V it's 1.8k and the next higher standard value would be 2.2k.

That'll limit the short inrush current even if all fets switch at the same time. The drawback is a slight delay because the series resistor and the gate capacitance are a low-pass filter. I would expect this to be neglible for your application because flipping the mechanical dot is likely to be several magnitudes slower than the electrical delay.