Welcome to Seven Forums rut. Good troubleshooting and info you gave us. The "trick" of trying a bootable CD is to see if the hardware (the optical drive itself) is at fault. The fact that the drive recognized and booted from the CD tends to point towards a software issue.
Do you have a dedicated program to play your media? What do the DVD's that work open in? You can try a clean boot
to see if the CDs are now recognized. Then enable programs a few at a time until you find what is interfering with the CD playback. The 1st suspect would be a media player and/or codecs.
Try the clean boot and let us know if this allows you to read CDs. As an aside, try a music CD as well, see if it is a data CD problem only.