Complementary pair - Role of capacitor on emitter
I am struggling to understand the role of capacitor C2 on the emitter
branch in the first stage

If I redrew it like this (where capacitor C2 is removed), does it make more sense: -

In other words: if you swapped the positions of C2 and R3 in the original circuit, you would see a DC voltage of about 9 volts across C2 (half of the 18 volts DC supply).
Q1 is acting as a differential input device where the base is the input signal and the emitter is where the feedback signal goes. A more sophisticated circuit might use a differential pair amplifier instead but, this will work reasonably well.