What's the use of the bypass capacitor on the LM380 amplifier?
First of all, the output coupling capacitor will never go negative.  The speaker will, but the capacitor won't, and that's one of the reasons it exists at all, to provide a dc offset so that the amplifier doesn't have to go negative in order to drive the speaker negatively with respect to ground.  It needs to be large enough so that it doesn't get charged/discharded too much during voltage swings, which would distort the audio signal.

Secondly, the datasheet includes a schematic on page 2, and you can see exactly where the bypass capacitor goes.  I'm no audio expert, but I can see that it is basically part of an RC low pass filter to power/bias the differential stage of the amplifier, so it basically isolates it from power supply fluctuations.  A side effect is to partially separate the dc bias current setting resistor from the open loop differential gain setting resistor of that stage.