I agree that this sounds like a Windows issue, since the drive will boot the OS disc and allow you to work from it. Since the drive itself does not show up in Device Manager, you might try removing the IDE controllers and reboot to let Windows redetect them. I've seen that work in the past with similar problems.
If you're unfamiliar with what I'm talking about, in the Device Manager you should see an item labeled IDE ATA/ATAPI controllers. Hit the plus sign beside that and under it you should see a Primary and a Secondary IDE Channel. Usually, the Secondary channel is the one the DVD/CD drive is on. Rright click and choose to uninstall that item, then reboot the computer and let Windows redetect it. If that doesn't work you can try removing the Primary channel as well, just don't be alarmed if it tells you that you must immediately reboot. Do so and it should redetect it as well. At that point, if it worked there should at least be an entry in Device Manager for the CD/DVD drive again.
If removing the IDE Channels doesn't help, you might try a repair install of Windows, since the CD is working alright. It could be a corrupted registry key as mentioned previously, and the repair install could well resolve that.
If all else fails, you can try the manual registry key removal route, but I'd consider this a last course of action. Here is Microsoft's page on removing the registry entries that relate to the DVD/CD drive: http://support.microsoft.com/kb/314060/en-us
I hope some of that may help.