Electronic – Convert the input of an ATX power supply from 110 V AC to 11-14 V DC

The ATX power supply is a switching power supply. It's important to be sure of the difference - a switching power supply switches the voltage at a much higher frequency (e.g. 65kHz) so smaller transformers can be used (transformers work more easily at higher frequencies) The downside is high current switching can easily cause EMI problems if not dealt with properly, and often in cheap power supplies (e.g. ATX) the filtering is very basic and may not always meet regulations.

A "standard" supply uses the 50Hz or 60Hz mains frequency to change the voltage, so the transformer for the same amount of power would be much larger. The supply size will likely be over twice as big for the same rated output. However with low frequency and (usually) linear regulation, these supplies are usually quieter. Are you absolutely sure your "AC/DC" supply is a switching supply?

It is relatively easy to make a quiet supply using either design though, so you may find a well made SMPS (can include linear regulation after initial switch regulation) is quieter than a cheap (possibly unregulated) standard supply.

I think your noise may well be present all the time, and causing issues with the sensitive (active I assume :-) ) analogue speaker circuitry. It may be (as mentioned previously by Madmanguruman) from the PC/monitor plugged causing problems with regulation, or could be a grounding problem of some sort. Depending on the frequency (be good to know roughly how high/low it is) it would be easier to guess at what is causing it. An scope on the lines would tell the full story.

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.