Grounding the chassis with capacitor
After some experimentation, I found out that it is indeed okay to have capacitative coupling between chassis and earth ground. In the question, I wrote, 


  if I connect the earth ground DIRECTLY to the chassis, the circuit is shorted.


This is not true, and it was actually caused by me shorting the 5V supply accidentally.

There were several problems at hand. First, my DC voltage supply was isolated and floating off some weird value. This meant that the 'negative' of the supply wasn't actually connected to the earth ground. As stated in the question, I wanted to drive a motor that was connected to earth ground. Fortunately, the motor only required a differential voltage, hence it functioned okay. 

Another issue was the microcontroller resetting whenever a large machinery turned on. This was solved by putting bypass capacitors between all VCC and GND connections (even for the LCD screen that I have as well). Thanks to @Horta for the suggestion!
Although, it still is a mystery to me why there was voltage fluctuation within the earth ground in the first place.

Thank you everyone for your help!