Decreasing Sample Rate in DFT (FFT) for Audio Analysis
I've read that to increase frequency resolution of FFT results one should dicrease sampling rate and increase window size (number of samples).


To increase frequency resolution, you increase the window size (increase the number of samples).  The sample rate does not change.

For simplicity, I'll pretend your signal is sampled at 800 Hz.

If you were to take 4 samples of the audio, and run an FFT on it, you would get 4 samples in the frequency domain.  They would represent 4 frequency bins: 

[0 Hz (DC), 200 Hz, ±400 Hz (Nyquist), -200 Hz]


Each of these bins is 200 Hz wide.  If instead, you had taken 8 samples of the audio, and run an FFT on it, you would get 8 samples in the frequency domain, representing 8 frequency bins:

[0, +100, +200, +300, ±400, -300, -200, -100]


Each of these bins is 100 Hz wide.  Does that make sense?  So to get narrower bins, you need to take more samples of your audio.  But the maximum frequency in the FFT (the Nyquist frequency of ±400 Hz) does not change; it's the same regardless of the number of samples.

Note that you can get these values like this:

scipy.fftpack.fftfreq(4, 1/800)
Out[8]: array([   0.,  200., -400., -200.])

scipy.fftpack.fftfreq(8, 1/800)
Out[9]: array([   0.,  100.,  200.,  300., -400., -300., -200., -100.])