Electronic – Driver transistor – what it is for

Oops. I got the order wrong in my original explanation, and have edited to reflect this. My apologies.

In your picture, Q5 and Q6 are being used as zener diodes. Current flows through the CB junction, which behaves as a reverse-biased diode. Since an NPN transistor has two PN junctions back-to-back, with the base grounded it looks like

^{simulate this circuit – Schematic created using CircuitLab}

Note that you ordinarily avoid exceeding the voltages where a transistor junction will act like a zener, but that doesn't mean you can't do it when you want to.

The other configuration you show, consisting of 2 transistors with common bases and the emitter of one tied to the base, is widely used in IC design, and is called a current mirror https://en.wikipedia.org/wiki/Current_mirror.

In effect, the first transistor acts as a diode whose voltage (the base-emitter voltage) is controlled by the current forced through it by the collector resistor. If the two transistors are perfectly matched and thermally maintained at the same temperature, a pretty good description of adjacent transistors on an IC die, the current through the second transistor will be forced to equal the current in the first, since both have identical Vbe characteristics.

1. By setting the output voltage of the circuit to about half the power supply voltage, the output can swing up about equally as much as it can swing down. For many simple input signals (eg. audio), this usually what you want. In practice you want to keep some headroom to prevent distortion.
2. The reason for choosing 5mA is in some sense similar to #1. It means in "worst case scenario" the input signal can reduce the current through the transistor by 5mA, resulting in 0mA. Again with the audio signal as an example this would mean a maximum current of 5+5=10mA. This is somewhat theoretical, in practice you will want some headroom to prevent distortion. The choice highly depends on the load you want to drive and of course the capabilities of the transistor.
3. Hard to say how to size the circuit for an 8 ohm speaker, it largely depends on input amplitude of the signal. I would personally use a different architecture for such a low impedance load. You could try your luck with an output series capacitor in the order of 220µF (+ goes at emitter, speaker goes to ground). When driving an el-cheapo speaker and being tied to this circuit, I would personally probably just try my own luck by putting the speaker in the collector lead.