DVD player power supply circuit strange behavior. Need help to understand root cause
Most likely there's nothing severely wrong with the power supply.

Your description leads me to believe that the power supply has its feedback on the +5V rail and having zero load on the +12V rail causes it to drift high. In the application (the DVD player) there's always load on the +12V rail which keeps the output closer to +12V.

High-pitch acoustic noise often comes from ceramic capacitors or the magnetic components. A faint noise is generally not a cause for concern - it was most likely always there but when it was enclosed you couldn't hear it.

That being said, it's possible that the power supply can no longer deliver its rated current, or the rest of the hardware is drawing too much current and forcing the power supply to protect itself. You need to check for short-circuits and dried-out electrolytic capacitors on both the power supply and in the rest of the DVD player itself, as well as probe any diodes and transistors you can find. ICs are tricky (especially if the unit is using some custom ASIC).

(Realistically, the way consumer electronics are designed these days, eight years is a remarkably-good run for a device that most likely only came with a 1-year warranty.)