possibilty of extracting a tone from MIDI values
You correctly note that a musical note is usually not just a simple sine wave. Thus, an FFT may only be one of the steps used in a pitch detection/estimation algorithm (there are many: http://en.m.wikipedia.org/wiki/Pitch_detection).  For music, a good place to start might be the research papers published on MIREX: http://www.music-ir.org/mirex/wiki/Audio_Melody_Extraction