Beginner A-class audio amplifier
If I remove the C2 capacitor I have no output signal.


C2 is your input coupling capacitor meaning it allows your AC signal to pass thru and not any DC current. If you connect your source directly to the rest of the circuit (direct coupling) without C2, you will effectively be shorting the base to ground thru your source and the transistor will no longer be in forward-active mode, which is why you get no signal.


  So why I was able to achieve only tiny gain?


This circuit doesn't stand a chance of driving an 8 ohm speaker. You are going to see extreme loading effects. For starters, output impedance is too high and it can't provide enough current to the load. You will need a power amp to drive a speaker. A push-pull stage might work if you design it properly.


  I am a bit surprised that it works at all, since audio input signal has positive as well as negative components.


The purpose of your resistive divider at the base is to bias the base above ground at a certain DC voltage. Coupling thru C2 allows you to superimpose your AC signal on top of that DC bias voltage. So you have a little room to swing negative. Might make it more clear if you look at the AC equivalent circuit.