In analog-to-digital conversion, how independent is the Nyquist rate from the power of the signal?
Completely independent. It is dependent on frequency only. The amplitude of the signal is only relevant when looking at the number of bits used (and the dynamic range).
See here for more details.
Theoretically, the rate is independant.
In reality, exceeding certain power levels will cause real world digitizers to increase in non-linearities (clipping, saturation, thermal damage, and etc.), which will introduce harmonics and other spectra above any Nyquist rate based on linear system assumptions, if not complete system failure.