To answer your question - yeh the speakers wires may well be mixed up at the back of the headunit but the popping sound when you turn the volume down sounds alarming! Does this happen only on the radio or when a CD is playing too?
Try playing a CD and see if the sound quality is improved, i suspect your ariel hasnt been wired properly. Sometimes they require a seperate earth!
I forgot to say earlier, on the sony ISO cable supplied with the radio there is usually bullet connectors between the Ignition live and permanent live to make it easy to switch between as some manufacturers have it the other way round! Should be easy to swap!
But sound quality wise, try a CD and see if its areial interference or incorrect wiring!
I've just put a CDX MP70 into my mates Accord Type-r and that really was a nightmare! The whole dash panel had to come out to remove the original crap!