I'm sure you know there is 120VAC at your outlet. The voltage is always there (assuming the outlet is not switched). Current (Amps) only flows when you plug something in and turn it on. If you plug-in two toasters or two hair dryers you'll get excess current, blow the circuit breaker, and the voltage will go to zero until you reset it.
And you might have some 240V outlets (with different sockets) for a clothes dryer, or electric range, etc.
What does that mean exactly?
If the voltage is too high you'll get arcing ("sparking") through the insulation between the conductors, or out to your body if you were touching it, etc. In reality, you probably wouldn't get that with 1000V, but the regulatory/rating agencies & standards are "careful" & "conservative".
In theory, you could have excess current through the cable. In that case it would overheat (usually at the plugs/contacts where the resistance is the highest) and it can start melting or burning and once the insulation melts it can short which would "pull" more current and blow the breaker. By that time it's too late.
But, any detachable power cord will be rated for the full 15A of a standard wall outlet so the power cord is not the weak link.
The gauge does affect current carrying capacity, but again,
any cable will as much current as the socket can put-out without blowing the breaker.
The resistance in the cable is directly related to the gauge (wire diameter) and length. When current flows through resistance you get a voltage drop. A 100W light bulb at the end of a 100 ft, 16AWG, extension cord will probably get (nearly) full voltage. But if you add a toaster or hair dryer at the far-end, the voltage will probably drop enough to see the bulb dim slightly.
Since most amplifiers have regulated power supplies, the internal DC voltage usually remains constant over a wide range of input voltages. There are a lot of amplifiers (and other equipment) with "universal world-wide" power supplies designed to operate between 85 and 300V (or something like that).
One cord is 14 gauge and the other is 16 gauge. I can't remember which one goes to which amp.
Practically speaking, it doesn't matter. But the amplifiers should be marked on the back with the voltage and the current and/or wattage. (This is not the same as the output power to the speakers because no amp 100% efficient and as you probably know, class-A is the least efficient.)
You can use the lower gauge (thicker conductors) on the amp with the highest power or current requirements. Power (Watts) is calculated as Voltage x Current if they are marked different from each other.