Electrical – How does a 700W PSU deliver so much power when most wall sockets are limited to about 20 amps

You need to rethink your calculations.

Let's assume that the amplifier you're using has negligible losses. Then the very greatest power you can provide to the speaker is (12 volts x 12 volts / 8 ohms), or 18 watts. Music, though, is essentially sinusoidal, so the maximum sustained AC power, technically called RMS power, is 0.707 times the peak power. This means that your amplifier can only produce 12 watts. Note that the data sheet you linked to suggests a 22 volt power supply. Checking this against the previous calculation gives (22 x 22 / 8) x .707, or 43 watts RMS. And using the absolute maximum of 24 volts gives 51 watts. Close enough.

So the peak current will be 12 / 8, or 1.5 amps per channel, for a total current of 3 amps, and a total power supply requirement of 36 watts.

So, to answer your question, the unit does not pose a fire hazard. It doesn't provide nearly as much power as you thought it would, but that's a different question.

As Passerby says, the main reason why there are several wires for any given voltage is that a computer may require tens of amps, so one wire by itself isn't enough.

The reason for all the voltages is largely historical. For many years, the standard voltage for digital logic circuits was 5V. Many circuits still run at 5V.

However, later chip designs could be made to work at lower voltages, and would run cooler when powered at a lower voltage. So 3.3V was added to the power supply.

+12V is mainly used for powering motorized things, like disk drives and fans. It's also needed for RS232 serial ports. RS232 also requires -12V, so power supplies often have a low-current -12V line as well.

The 5Vsb line is a "standby" power line, to keep some parts of the computer powered on while the computer is in "sleep" mode.