The remote start system WILL shut off if you get into the car and hit the brake. I've seen some Ford vehicles that have been started using remote start shut off when you open the door (Flex/edge do that alot. i think its because i use the key fob to unlock the doors instead of putting the key into the doorlock).
If you stop at the corner store and run inside, the car will continue running. If the keyfob is out of range from the SHO, the car will still run/shift out of park. If someone steals your car, they can keep going until it runs out of gas. But they wouldnt be able to start it after. So if you plan on leaving it run as you go somewhere else, keep it locked. I don't know whether or not the anti theft system will recognize a break in while its running and shut the car down.
Trying starting a car thats push button start (SHO/limited taurus, mks, mkt) when you have two or more fobs with you lol. I would literally have to hold the proper fob near the push button while holding all the others in my hand out the window of the car to be able to start it.