1.It does not explain how time is relevant to a wave being monochromatic; so why does the fact that a physical waves do not extend back to t=−∞ mean that physical monochromatic waves are an impossibility?

One has to understand what a mathematical model of a physical observable, in this case light of a certain frequency, means. It means that for the model to hold all its implications are manifest.

This means that the monochromatic model above says that if we go one kilometer away from the beam (lets suppose there is a monochromatic beam) the same intensity will be found ( lets alone what happens in time, that the beam should always exist, and we could always measure it). This does not fit our observations, because we have beams of light that start appearing , and stop appearing. BUT the model above is not useless, the mathematics leads to wave packets, which can have close enough frequency for our observations to apply "monochromaticity". Wavepackets solve the same wave equations and take away the problem of monochromaticity.

2. As relating to 1, how do the equations of the idealized harmonic waves imply that the wave exists for all time? Time t is an independent variable of the equations, which is free to be selected by us, so I'm a little confused about this.

It is a logical conclusion: if it is time independent it should be measurable at any time t, which as I argue above , is not what is observed

So we use the solutions of the wave equations so as to fit what we are really observing.

Quantum mechanics, which mathematically describes how light appears, also explains the physics of why there is always a width to the frequencies of monochromatic light: light comes from energy levels which have a width, which you will leaen if you study further in physics.

1.It does not explain how time is relevant to a wave being monochromatic; so why does the fact that a physical waves do not extend back to t=−∞ mean that physical monochromatic waves are an impossibility?

One has to understand what a mathematical model of a physical observable, in this case light of a certain frequency, means. It means that for the model to hold all its implications are manifest.

This means that the monochromatic model above says that if we go one kilometer away from the beam (lets suppose there is a monochromatic beam) the same intensity will be found ( lets alone what happens in time, that the beam should always exist, and we could always measure it). This does not fit our observations, because we have beams of light that start appearing , and stop appearing. BUT the model above is not useless, the mathematics leads to wave packets, which can have close enough frequency for our observations to apply "monochromaticity". Wavepackets solve the same wave equations and take away the problem of monochromaticity.

2. As relating to 1, how do the equations of the idealized harmonic waves imply that the wave exists for all time? Time t is an independent variable of the equations, which is free to be selected by us, so I'm a little confused about this.

It is a logical conclusion: if it is time independent it should be measurable at any time t, which as I argue above , is not what is observed

So we use the solutions of the wave equations so as to fit what we are really observing.

Quantum mechanics, which mathematically describes how light appears, also explains the physics of why there is always a width to the frequencies of monochromatic light: light comes from energy levels which have a width, which you will learn if you study further in physics.