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This may be a better question for https://physics.stackexchange.com/ but I'll give it a shot. I assume you show the FFT of the time waveform at the sensor. In order to get the actual frequency response you would need to divide by the FFT of the hammer response. Since the hammer is reasonably smooth, we can eyeball this. It also looks like your measurement ...

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"integrator is unstable since it has a pole on the unit circle" -- not true. This Z-1 is done all the time in CIC resamplers. The pole is --on-- the unit circle exactly, and is therefore stable. Fun fact, if you use floating point math this falls apart and the integrator blows up bc the value is not exactly 1, its 1.000000000001 (or something larger than 1)

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This is a very complicated problems and I don't think there exists a one-size-fits-all solution. You can try Matlab's $invfreqz()$ and see if it works for your purposes https://www.mathworks.com/help/signal/ref/invfreqz.html In general this is a error minimization problem but the actual data and the way you set the your error function and the search ...

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The cut-off frequency is whatever you define it to be. One standard definition would be the frequency $\omega_c$ at which the attenuation is $3$dB, i.e., $\big|H(\omega_c)\big|=\frac{1}{\sqrt{2}}$. As far as we know there is no analytical formula for the exact computation of the $3$-dB cut-off frequency of a moving average filter. More than you ever might ...

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