Ideally I would like to have an accuracy of +- one millionth of a second in this system
You might get that resolution measuring a brisant shock-wave, but it is way beyond the capability of a standard microphone. You need a sensor the size of 300um or so and the sound wave would need to have significant energy upto 150kHz to get a nice sharp rising edge on that timescale.
Measuring the phase difference of a steady sinusoid perhaps you'd get that accuracy for 20kHz, but again you'd need very careful microphone geometry and no acoustic reflections to confuse the reading, and this is a steady-state reading only.