Recently I had the experience of fixing a faulty DVD drive in an original Xbox 360. The drive would scratch any discs inserted, grinding a perfect circle into them. It would also refuse to open and when it did, it would repeatedly open and close. Both are common problems with the original model, and there are surprisingly simple fixes for both. Credit goes to Llamma.com and YouTube user, Frankieseshy.
Fixing the Xbox 360 disc grinding issue
What happens when the Xbox begins to grind a perfect circle in your discs, it's because the disc is spinning off-kilter and making contact with assembly for the laser. The issue typically arises when moving the Xbox while it is running, but sometimes (in my case especially), it happens any time a disc is inserted.
The solution is to apply foam pads to the interior of the DVD drive. When applied to the topside of the case, the spinning disc will make contact with the pads before it can make contact with the laser. Llamma.com sells the pads fairly cheap, and the repair takes a little over a half hour total.
Disassembling the Xbox 360
Applying the scratch protection pads
Fixing the Xbox 360 stuck disc tray issue
Over time, the rubber band that makes the gears inside the Xbox spin when the eject button is pushed, wears out. The trick to fixing the issue is either replacing the rubber band or strengthening it. Strengthening it is a simple process of removing the rubber band and submerging it in very hot water for 15-20 seconds and then immediately submerging it in cold water. This tightens the band back up and resolves the issue.
YouTube user, Frankieseshy has a fantastic four part tutorial on disassembling the Xbox and treating the rubber band.
These were the two resources that I used to fix the problems listed. Hopefully, they take care of your issues too.