You have a SATA Drive (which is irrelevant to the current issue as far as I can see), you have 2 PATA HDs and 2 ATAPI Drives one a DVD-ROM and the other the problematic CD-RW, right?

This is correct.

Your mobo presumably has 2 IDE connectors, so 2 PATA HDDs and the 2 ATAPI drives would be using all available connectors/channels but how are they connected? Did you connect the 2 HD as Master and Slave on the same ribbon and the 2 ATAPIs as Master and Slave on their own ribbon connection?

The 2 PATA's are connected on the primary as master / slave on one ribbon. The DVD & CD R/W are connected on the secondary as master / slave on one ribbon.

Needless to say, check the other drive that is sharing the same connection with the problem CD-RW drive to make sure their jumpers are set correctly, if one is set to 'Cable Select' so should the other one be set to. If 1 is Master the other has to be the Slave drive. A burner type drive should preferably be set as the 'Master' if the other is a plane ROM (Read Only).

I have checked the jumpers on both CD drives & they are correct. I have tried them both as masters to the other's slaves with the same results. I have tried the burner alone, configured as the master, but the BIOS still won't detect it.

I'd suggest start leaving the CD-RW Drive alone on the cable, using both jumper setting and also make sure the BIOS sees the drive. No point in booting to Windows if BIOS can not even see an IDE or ATAPI Drive first.
Make sure the ribbon cable and its connection you are attaching to the Drive is in working condition as well as the drive itself (Drive opens/closes, etc.).
When you unplug the ribbon cable from the back of the drive make sure all applicable pins are intact. Often people rip a pin or two when improperly disconnecting a tight ribbon cable from the back of a drive (CD and Floppy drive especially fall pray to this unfortunate incident more often than HDs).

I have new ribbons, & have tried the old ones as well. The burner door does not open/close. The light on the front does operate, however. I have removed the drives a number of times, looking not only at the jumpers, but the pins as well. I also have tried other power taps on the burner.

Disconnect the HDs as well if you must, leave only the problematic drive connected and boot to BIOS configuration screen, your goal at this point must be to get the BIOS to see the drive by using any connector, cable combination so taking the already working drives out of the setup would simplify the troubleshooting of the problem.

If I remove all other drives, should I hook the burner on the primary? Can I hook one PATA HD as a master & a CD drive as a slave (one on primary, one on secondary)? I'm thinking no, but it can't hurt to ask. Doe the fact that the door doesn't operate be taken as a sign that it has crapped out? The drive is 6 yrs old. (I know )

Make sure the BIOS settings are correct as well, if you messed around with it too much or unsure what a related settings should be, reset the settings first, reboot and try. Resetting the BIOS to its 'Fail-Safe' configuration could correct a setting that could be causing this problem which you might have overlooked.

I have tried resetting the BIOS to optimal settings (an option using f9).

If the mobo is older than a few months, checking the vendor's web site for BIOS updates could be helpful as well.

I'll have a look.

Note: In a desirable setup, the most accessed/used drives should be on Master position and left alone on its cable. If sharing the cable is a must have such as in your situation, the Slaved drive should be a Drive that is not as often used as the Master. HDs are generally prefered to be set as Master unless the HD in question is occasionally used.