It _shouldn”t_ hurt to connect the DC ground to the earth ground, but I don’t think you should. I don’t see how that would help.
I _think_ the jack (round connector) on the mega is the same as the Vcc (12VDC) on the RAMPS board. You don’t want to connect that to 5VDC.
What they are saying is that you can connect the USB to 5V and the Vcc to 12V, that will make the arduino and the screen get powered from the USB and not the regulator on the arduino.
I don’t think that was your issue either though, because it failed when you plugged it into the computer…