Is there any reason why I shouldn't just reduce it to 0 secs?

I don't know if zero means "always on standby" or "never on standby". Never tried it.

Give it a shot, let us know how it does for you. Personally, I like to be able to shut off the ignition but then still be able to turn the volume knob to wake up the player so that I can listen to it without the key in the ignition. I think mine is at 5 seconds or something.

Also, what's the significance of the battery icon on the unit?

It means the voltage has dipped to a certain level. I think the battery icon appears when the voltage drops to something like 10v.

It doesn't necessarily indicate something is wrong, because there are times you would expect your car's voltage to dip. Starting the engine, for instance, in some cars will do this.