When I turn off the ignition, the unit switches off but I can still see the standby light is on. Worse still when I press play, the unit springs back to life with music thumping out of the speakers.

This is a feature. When you turn ignition off, the unit is still powered through permanent power (there are two, permanant and 'switched', which follows ignition; the unit is actually powered through permenent one, other is just a signal). It goes to sleep (blinking stand-by LED), but gives you a chance to turn it back on and listen to some music while waiting for your wife to return form... wherever. If you fail to turn the empeg on after emplode-configurable interval (I think that default is 30s, or perhaps 120s), it turns itself off for good (no stand-by light), and you can turn it on again only by switching the ignition on.

As far as battery icon is concerned, probably your voltage indeed is on the low side after cranking. Nothing to worry about as long as it stays on just for few seconds.