Electronic – Using superposition to solve circuit with one voltage source and a dependent source

I didn't read your whole question, which seemed to go out of its way to make a simple thing complicated. As I understand it, you have a capacitor, resistor, and LED all in series, and you want to know how things decay if the capacitor starts out initially charged up.

At first apporoximation, you can consider the LED a voltage source. That means the current will decay just as if the LED wasn't there and the cap was charged up to the LED voltage less than what it really was. This is now a simple R-C systems which follows a basic exponential decay with a time constant of RC, which it seems you already understand. The question of when the LED goes "off" then comes down to at what current you consider the brightness to be low enough to be off. This can vary a lot by the efficiency of the LED, ambient light level, and how obvious "on" is supposed to be. For example, if the cap is initially charged so that the initial current is 20 mA (a common maximum for LEDs) and you consider 1 mA the "off" level, then the on time will be the 95% decay time, which is 3.0 time constants.

As I said, this was the basic first approximation where the LED has a fixed voltage accross it. That will be largely true, but of course its voltage will drop with current somewhat. For practicle purposes, this is a small effect compared to the slop of deciding what current level "off" really is, unless that current is small, like less than a mA.

The key is to realize that there is no such thing as 0 volts in an absolute sense. Voltage is a measurement of difference in potential between two points. You can say that something is 0V with respect to something else, but you can't say that a conductor is at 0V without including a reference.

Consider a bird sitting on a high voltage power line. The power line and bird are both at 13,800V relative to the ground, but the bird is at 0V relative to the wire.

To answer your question about AC, the AC source provides a potential difference (voltage) between the two conductors. It alternates in the sense that sometimes the first conductor is at the higher potential, and sometimes the second conductor is at the higher potential. "Higher potential" is an absolute concept; current will always flow from the higher potential to the lower potential if you were to connect them with a wire.

For a floating AC source like you've depicted (or a transformer), there's simply no concept of 0V until you define one. And you can define that any way you want: call the bottom terminal 0V, then you're in your first scenario. Or call whichever terminal has the lower potential at any instant 0V, and you're in your second scenario. Or define 0V as the "midpoint" when the top and bottom are at the same potential, and now you're in your third scenario.

When you use words like "stationary" you need to define what they're in reference to. Certainly the hot line does not stay "stationary" with respect to the neutral line, nor vice versa. In practice, it's common to use the earth ground as the reference point. In a house, it's common to physically connect neutral to the earth ground. Then, it's generally the case (modulo effects like resistance in wires) that the neutral will be at a potential of 0V relative to earth ground.

It sounds like you understand the concept of a floating DC supply, where there is no intrinsic reference to ground and you can choose to connect ground to either terminal. Extending this understanding to AC should be straightforward: consider that an AC supply is just like a DC supply where someone's constantly adjusting the knob, except that you can also bring the voltage negative (e.g. bring the black terminal to a higher potential than the red terminal).