Overheating Arduino Nano - power supply/capacitor issue?

Correct.
A double ground path is almost never a problem, but that depends on how the wires are tied together for example. Suppose the GND wire for the high current gets loose, then the large ground current would go through the Arduino board. It is better to be safe.
But I would also add protection resistors in the signal paths of the PWM signals to the Picobuck modules.

If the onboard voltage regulator gets hot (because of the 12V), then you can think of something else.
The voltage regulator gets too hot if you can not keep your finger on it.