How does one choose the proper windowing function to apply in an FFT? What are the benefits of larger sizes and smaller ones? Which would be the best for pitch estimation? Also which window function would be best for pitch estimation?
Depending on the pitch source (voice? Instrument?), the magnitude peak from a windowed FFT may not even be good for pitch estimation. If you are post processing FFTs as a sub-component of a more complicated pitch estimation method, then the type of post processing (parabolic log magnitude interpolation, complex Sinc interpolation, phase vocoder estimation, cepstrum, etc.) may alter the choice of more optimal windows.
A shorter FFT window provides better temporal localization for changing pitches. A longer FFT window provides more frequency resolution, providing the spectrum stays stationary for the duration of the window width.