What is the purpose of R6 and R7 in Grove sound sensor?
Here is my interpretation.

As you suspect C3 ac couples the output of the microphone, which is biased up about 1V when all is quiet, down to ground on pin 5 of U1B. 

When sound happens it will force the voltage on R6 up and down around zero volts..

R7 presents a higher input impedance to the op-amp and also prevents those negative noise spikes from back biasing the op-amp. 

Two amps are used, probably to provide enough gain at the frequency range required. There is sufficient gain here to cause the output to saturate with noise over a certain level.

C2. I have no idea why they decoupled the output, since that will depend on what comes next. 

This circuit is not suitable for capturing the sound itself for a few reasons, 


The low going swing on the mic is cut off by the zero volt bias on U1B
The gain is sufficient to swamp / distort the signal at high sound levels
Since the signal is biased to the rail of the opamp, again distortion is generated.


It is truly set up as a sensor, through perhaps not a very elegant one, not a microphone.