How can I use audio signal to actuate relay (or close dry contact)?
Detecting audio and making a digital line wiggle is pretty easy:



This circuit keeps Q2 not quite turning on.  Just a little signal thru C1 causes it to turn on at least for part of each cycle, which makes OUT pulse low.  OUT will swing nearly rail to rail and will be a clipped digital signal.  Now that you have a digital signal that goes low regularly when audio is present, you can feed this into a retriggerable one-shot and have the output of that drive a relay or whatever.