I think my car is possessed? But seriously?

I've never had any problems with it, really. Just an oil leak and an occasional flickering clock light. But last night my family and I were sitting outside and all of a sudden my alarm started blaring. The alarm on my car doesn't ever go off, unless you reach in through a window and unlock a locked door. I turned it off, and when I went back about an hour later all the dash lights were on. The car hesitated to crank and when it finally did, I turned it off and took the key out, but my car was still running! With the ignition turned off and the key out! I drove it to my house, and the engine went off but all the dash lights were on and the key signal was beeping to alert me I had left the key in the ignition, but I hadn't. A few seconds later it shut off. I went inside and about 5 minutes later I heard the alarm going off yet again. I turned it off, called my father in law, went back outside and the dash lights were all on again. I unplugged the battery cable so it wouldn't go off any more in the night but now I'm stumped on what could be causing it. Help? It's a 1998 Honda Accord EX 6 Cylinder engine.

Added (1). I don't Really think my car is possessed. I realize I'm having real technical issues.

The alarm going off has me confused but for the engine that will not shut off and the dash lights I have a answer. It may have something to do with the alarm I'm just not sure. There are many relays in the car which you can find there location by looking in the owners manual. One of the relays are sticking closed and it is allowing a current back feed into the electric system keeping thing running. It may be hard to find which one is causing the back feed because you need to pull it out when you have the problem and you could pull one that is not bad but will shut the engine off. You should buy one new relay to match what is in the car and when the problem is there pull one at a time until the engine/lights shut off, replace that relay. If it never comes up again you got the right one.

Your car must be having hell lot of technical difficulties. I hav seen some vehicles having such problems like light flickering for no reason… The only thing is you haven't faced anything like that other than your car… Which makes you think your car is possessed.
When you disconnected car batteries.did any such thing happen again? I don't think so… Cause u basically took away the electric supply required by it. And thats d real proof of it being a technical glitch as if it was really possessed weird things would have continued.