FFT for real signals
I suggest you do some more reading about the Fourier transform, but the short answer to your question is yes you can calculate the FFT for an arbitrary vector and discard the imaginary part, but it will no longer represent the Fourier transform of that signal. You can then perform the inverse FFT on that purely real spectrum, but it will not give you the original signal back (since it was not the actual spectrum of that signal).

If your starting signal happens to have even symmetry, then the spectrum will be purely real anyway, so you will be able to invert the real part of the FT and get your signal back.