Prove that time-reverse yields inverse filter for deconvolution?
This is not a general result. It specifically holds, iff the Fourier transform of the signal is of unit magnitude. In that case you can write the signal in frequency domain as

$$S(\omega)=\exp(i \phi(\omega))$$

for a real function $\phi$. The time reversed signal (I'll ignore the constant time offset, it doesn't change anything relevant) is then the conjugate

$$\bar{S}(\omega)=\exp(-i \phi(\omega))$$

Convolving the two by multiplying the frequency domain representation gives

$$\exp(i\phi(\omega))\times \exp(-i\phi(\omega)) = 1$$

Which is the response of the identity system and demonstrates that the signal and its time reversed are the responses of inverses.

This result is only exact in this special case. Sweeps can belong to this class of signal, for example if they have been generated as an allpass filter response. You can also do that by constructing the sweep entirely in frequency domain.

A sweep that is synthesised in time domain using a modulated sinusoid is typically not of that form, but approximates it well. Especially near the endpoints of the frequency range you will get significant deviation from the unit response.