DVD/CD Drive no longer shows in Finder

The first thing to try is to test the problem in a new user account. If the DVDs show up in the Finder in a new user account, then you know the problem is somewhere in the original user account's Library: a corrupt or misconfigured preference, most likely.

If the problem still persists in a new user account, then you know it is at a System level: so either a bug in the OS itself, or some weird interaction with third-party software installed in /Library, or possibly a global preference that is similarly corrupt or misconfigured.

Going back to the user-level problem: if that is the case, then you need to 'hunt the thimble': find the file that's causing the problem. You can remove large swathes of files and folder from the User Library, test, and then restore them if nothing changes, or restore half of them if it does, and so on.

Likely candidates are user/Library/Preferences/com.apple.finder.plist, and com.apple.sidebarlists.plist in the same folder. You may need to look for other potentially relevant filenames.

If you can see the DVD on your desktop, you can simply drag the DVD icon back into Finder.

You can enable showing DVDs on Desktop in Finder settings:

Then drag and drop your DVD back: