I finally got this solved, after too many phone calls, and too many visits from Dell's contracted repair company.
In the end, the issue was in the board that controls the DVD drive. We ended up getting one shipped from Dell, taking apart the laptop, and replacing it.
The board I'm talking about is the one right under the glowing eject button, etc. above the keyboard to the right. We had a faulty board, that was sending the eject command repeatedly.
Hope this helps people!