audio
Earth is the 3rd prong on the AC plug. The chassis is connected to this for safety, and often other metal parts like input jacks are connected to the chassis and thus the 3rd prong.
and
The live part usually goes to a transformer (as does the neutral) and is usually rectified to produce a DC (rather than AC) voltage to power the gear in question.
II don't understand your question. It seems like you have already answered it. The metal string of a guitar moves through the magnetic field of the pickup, which induces a current in the wire that is wrapped around the magnet in the pickup. This wire is connected to the output jack, which connects to your gear via the cable.