The output of the FFT is a sequence of complex numbers. Each complex number is a pair consisting of a real-part and an imaginary-part, typically shown as a pair (re,im).
If you want to graph the power spectral density you'll graph 20*log10( sqrt( re^2 + im^2 ) ), using the first N/2 complex numbers of the FFT output, where N is the number of input samples to the FFT
The transform gives information about the frequency content over the past period, so it is analysed as a reflection, starting at the most recent sample and working backwards. You can compute a phase shift of cosine directly from real and imaginary parts in the spectrum.
The Fourier Transform is really indifferent about time domain or frequency domain, it just transforms one function to another, and it happens to be the case that pulses transform to sinusoids and vice versa.
It also shows that time delay translates to rotation of the phase spectrum. The position of a pulse in time domain can be accurately stored in complex frequency information