Electronic – Why the emitter is earthed in the Common-Emitter configuration of a Transistor

When we say that the base is "grounded" in a common-base amplifier, we're talking about the AC small-signal analysis of the circuit. The key concept is that there is no signal voltage on the base terminal.

This does not mean that there's no DC bias on that terminal, if it is required for the circuit to operate. But note that the base COULD be directly connected to ground if the emitter has a negative bias (assuming NPN transistor).

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.