Is it possible to pick up noise in an audio circuit through the ground / earth wire?
Yes, long ground connections can pick up noise, which means the ground in one place is a different voltage from the ground at another place.  For single-ended signals, this ground offset voltage usually appears as part of the signal.

There are two common strategies for avoiding this situation:

Use differential signals.  The actual signal is encoded as the difference between two signals that are driven oppositely from each other.  Ground offset then looks like a common mode signal, which can be largely ignored by the receiver.

Pay attention to how things are grounded.  Make sure all the grounds are tied back to one place without other connections to elsewhere, like the local power outlet ground.  This ground net is then connected to real ground in one place only.  Also avoid having deliberate current flow thru any ground connection.  There should be separate returns for power current.  Never use ground as a power return.