They do unlock, and the dashboard and shifter both show "P" (steadily, not like the switch is loose or anything). I'll start waiting longer before I shut it off to see if that prevents it.Do your doors unlock when you shift to Park? Mine do when this happens to me, so I think the switch is just fine and at least the door lock ECUs is fine. On my 2011 this seems to pop up every now and again, but then goes away. I do wiggle the shift lever and the message will disappear usually. I've kind of thought that it might be caused by shifting to Park and hitting the Power button to quickly.
VIN # B0985
As to the remote thing, the second time I shut it down, it didn't complain...I had the remote sitting on the right edge of the passenger seat, so maybe it was just in a bad location.