It is definitely NOT a Ford thing for the remote start to shut the engine off when you open the door. My Key or not, the remote start should start the vehicle and allow it to run for the programmed amount of time. The Superduty can be set to 5, 10 or 15 minutes of run time. If you want more than 15 minutes, do the remote start routine again before it times out and it will give you an additional 15 minutes of run time.
For the My Key issue, you may want to try turning the My Key feature off and see what happens. I know it caused several issues on my truck when I had it on.