02 PreRunner Mysteriously Starts Up and Runs on It's Own

2002 Tacoma PreRunner Red 3.4L V-6 auto 595k miles (original motor & transmission)
Got home from work the other night, put my truck in park, turned the key to off, and removed it from the ignition switch. As I opened the door about 3 seconds later, the truck cranks up and runs by itself WITHOUT A KEY! It was as if a ghost started up my truck. It idled, so I stuck the key back in the ignition and turned it to the ON position. I then turned it to the OFF position and it kept idling. I pulled the key out and stuck it back in again, then turned it to the ON position and OFF position. This time, it performed properly and shut down the engine. The next morning, it started up and drove fine for about 15 minutes. As I pulled away from a fast food drive thru, it started sputtering and was sluggish, like it was running out of gas (although tank was 3/4 full). I made it to a shady spot where it died. It would not turn over and the dash lights barely came on. The tow truck driver checked it before jumping and said the battery and alternator were going bad. He towed it to my house, where I replaced both items (Carquest honored the lifetime alternator they sold me 6 years ago). It now starts, runs smooth, and drives fine. But when cranking it, it misses and backfires a good bit unless you pump the gas pedal once or twice. Kind of reminds me of my grandparents' Ford pickup from the late 60's when you had to set the points when you changed spark plugs. It mildly started this misfire behavior about a month or so ago, but I really didn't give it much thought. At first, I suspected the ignition switch, but the key slides in & out smoothly and turns off & on without any sticking whatsoever. Now I'm thinking it's the key interlock solenoid. Do any of you have any experience with that part? Any other suggestions? Thanks.

 

I think I'd be looking for a remote start system that went screwy.

 

It’s haunted. The ghost owner apparently still want to be the owner….

 

Drop the knee panel and look at the ignition harness that comes off the back of the key see if it is melted, then follow it to see if there are any strange electronics installed inline like a remote start system as mentioned above.