>You don't record each "waves" (nor is it even possible)

Depends on what you mean by 'record'. Generally the air pressure is sampled instantaneously, but it can be broken down into frequency components before recording/perceiving. That's kind of how our ears work and in DSP you do an FFT (which you can then pass through an iFFT to synthesize the original signal).

It has been a while since I study DSP, so I'm not sure if FFT can mathematically recover all the original sine waves (FT can). But I don't think it is possible practically, at least.

Yes, if you use the FFT or any discrete FT, you can perfectly recreate the signal up to the Nyquist frequency. To perfectly recreate the whole signal, you'd need infinite Fourier coefficients.

Thanks