The pre-amp board is also designed to allow the removal of the separate battery for the sound module. If you just remove the separate battery and provide a voltage derived from the amp’s main battery without a pre-amp there will be lots of distortion all all volume levels, not just really high levels.
If you only have distortion with long wires or loud volumes, the pre-amp may help. There are many potential issues that can cause distortion and the pre-amp fixes several of them!
That’s tough to answer - It depends on how the pot is connected and also the characteristics of the amplifier. The output of the sound module is a high frequency differential PWM (way above audio frequencies) and can be very sensitive to long wire lengths. That signal running over long distances may be the source of the distortion. The pre-amp converts this into a more standard line level output that contains only audio frequencies, so should be less susceptible to distortion with long wire lengths. Just keep the wires between the sound module and the pre-amp fairly short for best results.