^ exactly this ^
Question 1: Is this your all's consensus about the cause?
this is as far as possible from the case
ford did not offer a remote start option FROM THE FACTORY on Expeditions in 2004. the car will not start itself.
so at some time, somebody has put a remote start system on the car. and something is causing it to activate.
Question 2: Will a local shop deal with this or is this a FoMoCo deal?
this depends on ifs a dealer remote start system. if it is then a dealer is the place you want to take it, they will be able to test it or figure out if it is just set up weird. if it is not a dealer system, the only thing a dealer will do is charge you to remove it and make repairs to the car (if needed) to get it back in OEM condition.
if taken to a good local shop, they could do the same as above and could probably take a look at it and figure out if it is a faulty system or a weird configuration.
if i was doing that, i would start either looking under the dash and check or by a little testing on the car to figure out exactly what activates it (there is a good chance its not as random as you think) some systems dont come with remotes and can be activated by the factory remote, there is two common ways this can be done. the first is with newer heavily can controled cars (not the case with the car in question), for certain new GM and chrysler cars, when you turn the key, the cars computer actually fully controls the starting process, a quick flip of the key and the car then cranks the motor as long as it needs to get the motor started. since these cars already start themselves, you can actually use a fraction of the parts and just add this feature to the factory computer.
the other way this can be done is almost all R/S systems have a programmable trigger wire that will activate the system from something else like another aftermarket alarm of a wire in the car. on very common use of this is to hook the activation wire up to the cars lock wire then programming it for a certain amount of pulses, in this case, lets say three pulses (can be more, or could be a quick pulse and a long pulse...) now every time you press the lock button on the OEM remote 3x the cars lock wire triggers the activation wire of the R/R system and the car will start. as a side effect of this, pressing the lock button on the inside of the door will have the same effect. pressing the last two buttons at the same time on fords with a drivers key pad, will lock the door, so pressing them 3x will also activate the starter.
if this is the case then the system is not failing but is working as setup.
As soon as you open the door and sit in the driver's seat, it shuts itself off. Sometimes, you have to put the key in the ignition, "start" it, put it into gear, put it back into park, and shut it down
this could also not be as random as you think.
most people automatically press there foot on the brake pedal as soon as they get into the car or very shortly after in the starting process, pressing your foot on the brake will always shut down a car that is remote started. so if you get into the car and step on the brake before you insert the key and turn it to the ignition/run position, it shuts off. but if you get in to the car insert the key and turn it to the ignition/run position then step on the brake the system turns off but the car will stay running because the key is on and is now keeping the car running, since you have to step on the brake to shift the car and drive it, the remote start system could remain active until you are trying to drive it.
I would also disassemble the under dash and find the system, see what model it is and download the manual and verify how it is hooked up to see what the real deal is.
, did a little looking on how Ford's system actually works. Yes, they are apples and oranges, but both of them are fruit. VATS and PATS are different in that one uses a resistor and the other uses an encoded chip, but both are pieces of crap.