How to take inverse fft of windowed and callibrated fft data?
When using quantized data formats, applying a window function is a lossy process, and arbitrary original data cannot be recovered, either before or after an FFT/IFFT process.  

You could try multiplying by the reciprocal of the window (being careful to avoid divide-by-zero), and get a noisy reconstruction (except perhaps for an end point or two).  And noisier at the ends, where the window tapers to near zero, thus potentially adding a huge amount of quantization noise.

Or you could overlap Hamming/Von_Hann windows by 50%, where the sum of the sequence of overlapped windows has a gain of 1.0 (except at the very ends), and thus is not lossy (or as lossy, but still with some amount of tiny numerical or rounding noise).