|"It happens for all original audio and software CDs."|
"For burned CDs behavior is inconsistent, either they are not recognized at all (no pop up, no files) or are recognized as blank (pop up, no files)."
So you have more besides legacy audio CDs that aren't being recognized . That sounds you're having problems reading ALL CDs.
If what follows this first part doesn't help....
"A "DVD-RAM" drive in Windows XP is a drive capable of burning and reading both DVDs and CDs.
It has at least two lasers and associated circuits. Sometimes one laser or the circuits for one malfunctions after the drive has been used a lot, but in that case, the drive would read only CDs or only DVDs."
...then I suspect the laser that reads CDs and/or it's circuits are malfunctioning, and you need to REPLACE the optical drive.
If you have a desktop computer, LG drives are inexpensive and quite good - relatively well balanced and less noisy when spinning a disk - but they're not the best for reading badly scratched disks.
Software problems would be consistent, not inconsistent.
If this is a laptop computer.....
Try cleaning the laser lens.
On most laptops, when the tray is ejected and no disk is on it, the laser lens is obvious on the top of the tray - use a tissue or soft cloth to clean it off.
If this is a desktop computer -
Try using a laser lens cleaning CD in the drive.
If you have more than one optical drive, try the disks in the other drive.
If that doesn't help...
There may be there may be a problem with the data cable to the drive
It is common to un-intentionally damage IDE data cables, especially while removing them - the 80 wire ones are more likely to be damaged. What usually happens is the cable is ripped at either edge and the wires there are either damaged or severed, often right at a connector or under it's cable clamp there, where it's hard to see - if a wire is severed but it's ends are touching, the connection is intermittent, rather than being reliable.
Another common thing is for the data cable to be separated from the connector contacts a bit after you have removed a cable - there should be no gap between the data cable and the connector - if there is press the cable against the connector to eliminate the gap.
80 wire data cables are also easily damaged at either edge if the cable is sharply creased at a fold in the cable.
Try another data cable if in doubt.
Check your SATA data cables. The connector on each end should "latch" into the socket on the drive and on the mboard, or on the drive controller card - it should not move when you merely brush your hand against it near the socket - if it does, mere vibration can cause a poor connection of it - use another SATA data cable that does "latch", or tape the connector in place.
(There is a slight projection or bump on one side of the outside of the connector that "latches" it into the socket - it's easily broken off or damaged)
The same thing applies for the SATA power connection.
For burned disks - "you should be able to read a CD-R (or DVD-R if the drive can read them) disk on any drive, but other types of burned disks may not read properly in a drive brand and model they were not made in."
That's mostly due to...
Not all disks will read properly in all optical drives.
All optical drives have specified lits of media - brands and types of burnable disks or types of readable disks - that will read or burn in the drive properly.
An original (a.k.a. factory, pressed ) disk - of a type the drive can read - should read fine in any drive. E.g. legacy audio CDs (1X, *.cda files), program installation disks, device installation disks.
In almost all cases, any brand of burned CD-R (or DVD-R if the drive can read them) can be read in any optical drive.
However, for other types of burnable disks, some brands of a type the drive can read will read fine in the drive, some won't.
If the brand is not listed on the drive manufacturer's web site for a particular type of burnable disk, there's no guarantee it will work properly in your drive model.
E.g. The only brand listed for LG optical drives for all types of burnable disks is Verbatim in North America, a.k.a. a different brand in some other countries, mostly Asian ones.
Some optical drive brands are pickier (e.g. Pioneer) regarding the media you can use in them than others (e.g. LG drives aren't as fussy).