I doubt the blower was designed to work when the key was in the Start position, so by not working in the Start position currently would not be a problem. Usually most of the accessories are shut off in the Start position to enable as much current as possible to be available for the starter motor.
But also the blower motor usually wouldn't work in the Accessory position, only the Run position. And since it doesn't shut off all the time when the key is turned Off, that does sound like an issue with wiring or an electronic module somewhere.
Did you change the ignition switch, or the key cylinder switch? A lot of people refer to the switch you put the ignition key in to as the ignition switch, but the ignition switch is a separate component activated by the key cylinder switch.
The issue with the door locks, dome light, and power windows sounds like either an issue where the accessory delay relay is not getting a signal that a door opened or like the Body Control Module (BCM) is malfunctioning. I know Chrysler has had a lot of issues with the BCM in the past....
-Rod