So this response in confusing...
Key on, doors closed, radio on:
yes radio turns on by itself - Normal
Turn Key off.
radio will continue to play for I think 15 minutes then turn itself off - Normal RAP
Do the interior lights turn on?
yes when doors open then go off after like 20 seconds - Normal
Open the door, does the radio power off?
yes - Normal
Open all other doors, does the radio turn off?
only tested driver door - K
Close all doors, but leave the drivers door open. With your FOB (assuming you have one)
yes, lock the vehicle by pressing twice on the lock button. Does the horn honk twice, or
once? - Not Normal.
With the door open, and you pressing the lock function on your FOB, it should honk twice to let you know that A Door is not closed.
That said... I think I misunderstood your original post. The radio is operating normally, but there is no way to actually turn it off.
So I reviewed the owners manual, and no where in there does it say how to turn it off. The closest I can get is holding the home button, but it appears that only turns off the display, and not the unit itself.
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SO I think your only option is to either figure out how to quickly access the mute button, or, wire in a switch to the red wire as you suggested above.