The easiest way to back up a game, is with wxripper and doing hot swap. I don´t even have to open the DVD reader (LG GSA-H10N) to do it, i use a clip to eject the disk. The only tools you need for this is Wxripper to exctract the data and Xbox SS Merger 1.6 to merge SS. Of course you neet to get the ss file for the game you are backing up, just google for them.
Finally I burn the image with DVD Decrypter et voilà, in less than one hour you have your backup made. SS Merger 1.6 is great because it lets you know if you have a valid image and SS match by comparing MD5 (users can share their experiences with each image, I think that the database is an invaluable part of SS Merger), so this way you have an extra insurance.
Use Verbatim disks, they can be burnt at 6X, so in less than 15 minutes you're done (vs. 45 minutes if you burn them at 2.5X).
If your DVD drive don't respond well to drive swapping with a clip, then you must work with the drive open (careful to be exposed to the laser!), in the end i think is better to have a dvd drive open than your x360 dissasembled...

Cheers and good luck!