To make sure that the Arduino Mega 2560 was ok, my partner connected the Arduino board to his computer using a standard USB cable. The power LED came on and glowed constantly. He tried to upload a basic sketch to make the Arduino blink. The uploading of the sketch, however, took forever and never ended.
Well, unfortunately that pretty much says your board is "not ok".
I'm guessing the motor shield you have is quite underpowered for the motors you are trying to run and one of the bridges did indeed overheat and break. A possible failure mode there is short-circuiting the 11.1V from the bridge voltage to the digital Arduino inputs which would indeed fry the Arduino. I'm not sure that's what happened but it's plausible.
The Rugged Motor Driver
: two H-bridges, more power than an L298, fully protected