Q: Start up without key in ignition

This has happened several times. My Vue will start up after I turn it off get out and use the remote door locks. After the cars horn beeps to confirm all the doors are locked the Vue will start up. At first it would keep running and it wouldnt even shut off when I put the key in turned to auxiliary then to off position. I would have to turn it all the way to the start position and grind the starter then turn to off for it to shut down. Now it doesn't to this every time just once in awhile. But yesterday was different I got out hit the door locks on remote the horn beeped I started to walk away Vue started I turned around holding the key in my hand start to walk back and it shut down, then started up again then again it shut down. I unlocked the doors with the remote opened the door and once more it started and then shut down. Now each time it started it ran for about 30 or more seconds then shut down.

My car has 100000 miles.
My car has an automatic transmission.