I believe the DVD drive is proprietary to the Xbox. Meaning you can't simply pull it out and stick it in your PC. The case is definitely different since it has feet protruding from the side of the case.
You definitely can't use the CPU in a PC since it's an Tri-Core IBM Xenon CPU which is based on the PowerPC CPU. And it is soldered into the motherboard.
My suggestion is to keep the old Xbox 360 for parts, like the DVD drive. However, swapping a broken DVD drive with another Xbox 360 DVD drive is not that simple. I believe the Xbox 360's motherboard contain unique identifiers for each component. Therefore, assuming the DVD drive failure is mechanical and not the circuit board, you need to transfer the circuit board to the replacement drive before installing it. Otherwise the Xbox 360 will not recognize the "new" drive.
Basically, the only part you can re-use is the 2.5" SATA hard drive (laptop hard drive).
The original XBox was just a bunch PC parts thrown together. Not the Xbox 360.
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EDIT:
BTW, I don't own a XBox 360 nor any other gaming console.