You are right, many of the "registry fix" tools are crapware. But there are some reputable tools that really do help fix registry problems. Your best bet is to look for reviews of these tools in reputable magazines to find one that works reasonably well.
However, I suspect that you'll find the registry so messed up with intersecting problems that no tool will be able to fix it.
Wait on buying the DVD drive until after you re-install Windows. As you suspect the problem may be with the driver or registry rather than the DVD Hardware.
Yes, having to re-install windoze is rather severe, but it has been a reality for the last 20 years. There have been people who advocate re-installing windows every 6-12 months, "whether it needs it or not". They tend to be techno-geeks who install and uninstall
a lot of software. However, every Windows user has to be ready at any point in time to re-install windows.
But, then again, simply re-installing Windows is a lot less drastic than going out and buying a new computer and stuffing the possibly repairable one in the closet.
Possible fix for updates.
First thing to do is make an image copy backup of the current situation. I assume you no longer need the data files, that they were all transferred to your new computer.
If you want, try the registry fix tools, run a diskchk scan.
Your best, only bet, is to go into Windows Update and look at the Update history. Go back to the oldest failed update and double click on it. There is often an error code you can google for and often a hyperlink to possible fixes. You can also post the details
of the failure to this site under Windows category to get help. After you apply the fix do a Windows Update scan and try to apply only the one patch that you tried to fix. If it installs, pick a few more of the oldest patches and select them to apply. Continue
applying the patches in small batches, rebooting periodically whether it needs to or not.
At 3 years old you are in the leading edge of possible disk failure. Disks either fail almost immediately after first running or will run for several years before dying.