Sounds to me like you may have a short in the starter solenoid, which is what engages with the flywheel at start up. You may have something going on in the electrical circuit that is diverting current without the key in the ignition. I had this same scenario happen to me, only I had a stick shift that was in gear without a parking brake, the vehicle started on its own and was moving down the driveway. I replaced the ignition switch for precautionary reasons and the starter and solenoid.
It might be best to have a technician check your electrical system. It could be why the battery is dead as well.
It sounds like a problem with the starter solenoid, but the ignition switch may also be playing a part in this as well.
〉 Answered on Aug 8th, 2008 by Lori Johnson, Owner and Instructor at Ladies Start Your Engines