What happens if I switch C and R
Yes, you can interchange their places. Nothing is wrong with that. It won't change or affect anything.

That capacitor is simply a bypass capacitor. The opamp runs in comparator mode and amplifies input signal. The amplified input signal is rectified on the bridge diode. That capacitor is the return path of the rectification process. It helps keeping the inverting input symmetrically swing around the middle voltage point (6V). It behaves as a jumper (short circuit) for the audio signal, however it behaves as open circuit for the DC. Actually, if the frequency was above 10MHz~100MHz levels, I would have suggested placing the capacitor as away as possible from the IC inputs for better noise immunity. But at audio frequencies, it just makes no difference. You can change places of R3 and C2 as you wish.