Is it accurate to say in digital audio that, when a fader is down, then its value is "-$\infty$"? Is it accurate to say in digital audio that, when a fader is down, then its value is "-$\infty$"?

What has confused me is, whether it's really $-\infty$ or some lower bound of the amplitude. it could be that the $-\infty$ is just a "joke" by the developers and it doesn't have mathematical validity.
If by "its" you mean the value of the fader, then yes - it's absolutely correct.
Fader defines the attenuation of the signal with respect to the reference level. The units are in logarithmic scale.

For example, no amplification (or in other words, multiplication of the signal by 1) corresponds to $0\; \mathrm{dB}$ since $10\log_{10} 1 = 0$.

If you want to completely attenuate your signal, then you have to multiply everything by $0$, hence $10\log_{10} 0 = -\infty$.