That's not what your schematic shows. I'd look up some schematics of high-side PNP switches if I were you. And after that I'd forget what you saw and just use a low-side N-MOSFET switch (or NPN if you want to go the 1970s way with this.) Don't forget the freewheel diode!!
In any case, don't try to run a fan the way you've connected it now. There'll be trouble.
As to your microphone sensor: that looks like a pretty standard run-off-the-mill comparator circuit using a microphone as its input. Probably one half of the opamp amplifies the signal and the other half acts as a comparator, with the pot setting the bias for the comparator. That would be my guess, but I'm too lazy to chart it out based on the product photos.
Anyway, my earlier advice still stands as to creating the right sound profile and selectively triggering on it.