Dangers of Old ATX Power Supplies?
Can't make out what the picture is supposed to be of, but the mush you refer to is likely some thermal compound or adhesive.
It's probably one of the inductors or transformers on the board, which can vibrate slightly when passing a switching current. This is called magnetostriction. Capacitors can also buzz but it is more likely to be the magnetics.
It is not an indicator that the power supply is faulty. You can dampen it by making sure the magnetic components are solidly fixed to the PCB (e.g. with the adhesive mentioned above)

I don't advise trying to service it yourself (no need if it's not faulty anyway) but just in case and since you asked: You probably want to wait at least ten minutes (the longer the better) after unplugging the supply, and then make sure no significant charge is left in the large capacitors. Most large caps will have a high value parallel "bleeder" resistor across them so they discharge slowly when power is turned off, but do not rely on one being present and working. Some capacitors can hold a charge for days if no means of discharge is present.
See the section "why this matters" and "discharge technique" on this page for some decent info. Be very cautious, you can receive a lethal shock from a large filter cap, or they can explode or even vaporise metal.