Why the microphone input is connected to the emitter of Q1 instead of
collector?
I think you might mean why is the microphone connected to the emitter instead of the base of Q1. It would be very unusual to use the collector as an input but, using the emitter is quite common. It's called a common-base circuit.
If you think about a standard NPN common-emitter circuit, the input is causing a signal current to pass into the base due to the emitter being held at a fixed DC voltage below the base. Well, in an NPN common-base circuit, the same happens (a signal current flows in the base-emitter) but, this is achieved by "wiggling" the emitter (with the signal) and holding the base at a fixed DC level above the emitter.
In other words, there are two ways to achieve the same thing. So, why not feed the signal directly to the base as in a common-emitter circuit you might ask. The main difference is that the input impedance in a common-base circuit is much lower and, this might suit certain microphone types (compared to a common-emitter circuit).
As for the rest of the circuit, the next stage (Q2) is a common-emitter followed by a common-collector. And, there's overall negative feedback from Q3's emitter to Q2's base to provide a good degree of stability.