Short answer: A solder joint came loose. Fixing it probably involves replacing a circuit board. No, your electronic wizard friend cannot find the break and resolder it. We are talking about a tiny solder joint on the pin of a chip. You might even need a magnifying glass to see the pin, and forget about seeing the crack.
Long answer: Electronic circuits are full of components that can resonate with each other due to their inherent (unwanted) capacitance and inductance. Even the traces of a circuit board can set up a resonance, like a wave guide. Electrical engineers carefully design filters to eliminate those resonances. Usually those filters are in the form of little capacitors, placed all over the boards, which sink stray signals to ground. These days, transistors are sometimes used for their inherent capacitance, because hundreds or thousands of them can be put on a chip, allowing for creative design solutions. They can also be cheaper than actual capacitors.
All of these filters are soldered onto the boards, just like all the other components. The chips tend to be very small, and since they carry almost no current, the leads can be made really tiny and close together. If one of those solder joints breaks, then an unwanted signal can create problems. If that signal is at the front end of a power amplifier, it can create the problem you are having.
Your speaker cone is not making the sound. It can't vibrate at that frequency. The sound is probably coming from the windings of your speaker coil, which are vibrating from this unwanted, amplified signal being pumped through them. It could also be from a vibrating wire or electrical component, but I would bet on the speaker coil. Either way, the vibration is not good, even if you can tolerate the sound. It will cause other things to fail earlier than they should. Also, if it is in the coil, that means your speaker windings are dissipating extra power that they were not intended to handle. That heats them up. The power rating of your speaker is partly based on how much heat the windings can dissipate.
The 10s decay that you noticed is just the power supply discharging. That has nothing to do with your problem. The first thing that happens with the electrical input is that it gets converted from AC to DC. There is a large capacitor that helps smooth the DC, and it takes a few seconds to discharge when you disconnect the power.
One last comment: Before you take the speaker back to the store, you might want to try plugging it in at a friend's house. It is very unlikely, but possible, that your speaker is picking up a stray signal from the environment or from your power line.