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Is it common for pieces to have lower notes played slower? Do instruments with lower registers typically play slower?

I ask because notes in lower octaves are closer together in frequency. For example, A0 to A#0 is about 1.6 Hz different while A1 to A#1 is about 3.3 Hz different - this is just a fundamental consequence of the doubling of frequencies between successive octaves.

At least with Fourier analysis, to distinguish two frequencies only different by 1.6 Hz you would need 1.25 seconds of the note being played. (that is 2*1/1.6 where the 2x is to handle the Nyquist frequency issue).

For A1 (delta of 3.3 Hz) you would need about 612 ms, for A2 (delta of 6.5 Hz) about 305 ms, for A3 (delta of 13 Hz) about 153 ms, ...

Is human perception able to distinguish notes better than this and thus not as limited by the speed of low notes as a Fourier Transform would be? Do lower notes usually get played slower to accommodate for this? With the logic above, A3 can only be played 6.5 bps or 392 bpm (eighth notes at 180 bpm).

The first thing I suggest you think about is the fact that a musical note is not a single frequency. Depending on the timbre of the instrument, there can be hundreds of frequencies present in a sound wave even when only one note is played on one instrument. We perceive the pitch of a note through our ears analyzing all of the frequencies simultaneously, not just the lowest frequency.

Also, human pitch perception is not accomplished by our ears or brains doing fast Fourier transforms. It’s an analog electromechanical analysis engine, and we have two of them. Our ears do have a sort of “integration time” if you will, and like an FFT, the integration time is shorter for higher frequencies, but again we are using a lot of midrange and high frequencies (often called “overtones” when perceiving the pitches of low notes.

Is it common for pieces to have lower notes played slower? Do instruments with lower registers typically play slower?

Yes, for a number of reasons. It often takes longer for a low-frequency resonator to settle into stable oscillation; It's often physically harder to change frequency quickly on a larger instrument (which tend to be the lower sounding ones); and the harmony of most styles of music tends towards slower motion in the bass voices.

I ask because notes in lower octaves are closer together in frequency. For example, A0 to A#0 is about 1.6 Hz different while A1 to A#1 is about 3.3 Hz different - this is just a fundamental consequence of the doubling of frequencies between successive octaves. At least with Fourier analysis, to distinguish two frequencies only different by 1.6 Hz you would need 1.25 seconds of the note being played. (that is 2*1/1.6 where the 2x is to handle the Nyquist frequency issue).

I'm not sure human pitch perception is quite as dependent on length of sound as you suggest there, as the cilia in the human ear can detect pitch components directly without needing to do a 'fourier analysis' as such. However it is true that we are less precise in our ability to distinguish the pitches of very low sounds in general. If you look up information on experiments to discern the pitch difference limen of the human ear you will find more information. If I recall correctly, the difference limen - the minimum difference we can hear - is around 1Hz through much of the audible range, becoming much bigger in the higher frequencies (where we are also less precise).

Todd's point about musical notes not only consisting of the fundamental frequency is also very relevant.