This generally doesn't happen , Since windows XP is suppose to load all these drivers by default.
You just wrote about one DVD drive but the Device manager shows up two . Is the second one a virtual drive or is it there physically ?
You should try removing the drivers manually (in Device manager > Right Click > uninstall) and then scan for hardware changes.
This will try reinstalling the device automatically. But If It asks you for any Drivers, I don't know what you would do

since both of your drives (according to Device manager) are not working
Lets just hope it picks up drivers automatically and installs them
Otherwise, You could go to the manufacturer's website and see if they have any specific drivers for their drives.
NOTE : I won't recommend you trying to upgrade your DVD-ROM's firmware right now. If it were the firmware's fault you won't be seeing it in the BIOS itself. Your OS (XP) could be the culprit, try running a live CD (Ubuntu) and see it works properly
Hope this helps