Windows Media Player often uses a different type of wiring to deliver sound from audio CD's to your speakers, often bypassing the need for a sound card completely. (If you are familiar with the internals of PC's, the CD AUDIO CABLE is usually a small grey cord that goes directly from the back of your CD ROM to a port on the sound card or motherboard. This basically acts as a conduit bypassing the sound card completely. This is why an audio CD may work but other sounds do not.
To be exactly sure what is going on, you need to test other windows sounds. Forget the audio CD's for a minute. Go find some MP3's or a .WAV and try to play it. Since your volume icon is gone now too, you will likely get an error if you try to play any other sound besides CD AUDIO.
Also, check your DEVICE MANAGER and tell us if everything in there looks ok or if anything has the dreaded yellow exclamation point or question mark. If there is an exclamation point or question mark, you can sometimes right click and try to UPDATE DRIVER for that device. It sometimes can download a replacement driver from the Windows Update location.
If the UPDATE DRIVER does not work (and you still have an exclamation point or question mark) then you may need to download the drivers directly from the manufacturer of your PC or sound card.
And incase you are not sure how to get to DEVICE MANAGER, you can right click on the MY COMPUTER icon and then go to MANAGE. Then there is a DEVICE MANAGER heading on the left you can click on. You could also right click MY COMPUTER and go to PROPERTIES, then click on the SYSTEM tab and then click the DEVICE MANAGER button.
I hope that may help!