Transistors – relation between impedance and gain

Input and Output impedance are all about power transfer. So there are two common ways of creating a circuit:

1. Bridged - generally this means that the output impedance of the last stage is transferring either voltage or current as efficiently as possible (that means that you're either transferring voltage and no current or current and no voltage). The power dissipated in bridged configuration is minimal, but the inverse is that you don't transfer the maximum power either. You also can suffer from reflections and other wave irregularities if the distance to wave-length ratio is too close.
2. Matched - this means that the input and output impedance of the stages are matched (go figure). The power dissipation is the highest and thus so is the power transfer (50%, yeah that's the max). With this you don't have to worry about reflections or wave irregularities as they are completely sunk into the load. This is one of the reasons that RF and microwave circuits use matched instead of bridged topologies.
3. The ugly middle - well, nothing's perfect. So there's always some trade off. No bridged topology every works perfectly. There's always some current transfer and some voltage transfer regardless of what you want to transfer. Matched lines in the real world can only be matched over a narrow bandwidth. Parasitic capacitance and parasitic inductance change the impedance of things enough that you start getting wave reflections and other irregularities (and that's just the transistor, then you have to deal with the resistors and the actual wires. Uhg. It can be a pain sometimes). Yep. Number 3 is the real world, in all it's glory.

The amplifiers you are speaking of are audio, that is, they only concern themselves with AC. As such, they are ordinarily AC coupled at both input and output with capacitors. So their DC input impedance is expected to be infinite, and their DC output impedance infinite.

EDIT - You added your schematic while I was composing this. Note the input and output capacitors. END EDIT

For amplifiers which are expected to include DC, DC input impedance must be specified, and likewise output impedance. You'll see both of those addressed in the data sheets of operational amplifiers. For inputs, you'll see specifications for input offset voltage, input offset current, and input bias current. For outputs, you'll commonly see the maximum voltage swing available for one or more load resistors, which is directly related to output impedance (although the presence of loop gain complicates the matter).