Does the inverse Fourier transform only produce stationary time signals? I have a question regarding the inverse Fourier transform and its relevance to non-stationary signals. And by non-stationary signal, I'm talking about a signals whose frequency content varies with time. I'll provide the example that made me originally ask this question.

Say for example I have a sinusoidal signal with a single frequency that goes on for ever. I could take the Fourier transform of this signal to obtain its frequency content. And because this signal is a single frequency that goes on forever, this signal is considered stationary. 

However, now let's say I have a signal that is zero except for in some window in which the signal is sinusoidal. This nonzero window exists from t = 0 to 1 second. If I were to take the Fourier Transform of this signal over a finite duration (such that it acts on both the nonzero and zero part of the signal), I would get a frequency content that shows there exists a peak frequency content due to the nonzero window.

Now let's say I were to do the same thing, except this time the nonzero window exists from t = 6 to 7 seconds. If I were to take the Fourier transform, over the same finite duration, I believe I would yield the exact same frequency content (both phase and magnitude) from the earlier experiment.

Assuming what I said was correct, it should thus be IMPOSSIBLE to recover the exact signal from the Fourier transform since two different time domain signals have mapped to the same frequency domain signal. Am I correct with this statement? Furthermore, I believe the inverse Fourier transform will end up producing a stationary signal. As in, the frequency that was once contained just within the nonzero window will now be shown to be infinite from the inverse Fourier transform.

TLDR: 


Does the inverse Fourier transform only produce stationary time signals?
Thus if I have non-stationary signal, take the Fourier transform and then take the inverse Fourier transform, I will now have a signal that is stationary?
The FT of a finite length sine wave burst consists of more than just the one frequency of that sine wave.  A finite width time domain signal has an infinite width FT.  Same for an FFT, a rectangular window produces a full width periodic Sinc in the FFT result.  So even if the phase of the 1 sine wave result bin stays constant, the phase of all the other frequencies will rotate as the position of the window changes.  Two different rectangular windows will NOT map to the same FFT, but convolve against any single sine wave differently.

Therefore an inverse FT or IFFT of the full spectrum can produce a signal whose statistics can vary significantly over different portions of the result, by reproducing a window which can be zero in parts of the result, and non zero in others.