My Demon is possessed

I was changing my alternator yesterday and when I had it off suddenly the car tried to start without key in ignition or me touching anything at all. I was a little worried about putting my hands back near the fan when I installed the new alternator, but luckily it didn't do it again. Today I went for a longer ride to test it and after half an hour I noticed a loud noise when shifting in neutral - something that I always do when I approach a red light or when Im standing. I left it in D until I'm at the house. When I put it in parking I immediately realized its the starter turning on in neutral and parking. I left it in reverse and turned it off and disconnected the battery. I also noticed that the positive on the battery was pretty hot and some acid seemed to be leaking from the top of the battery. Too hot maybe? Not sure if its related but is that a faulty starter relay or a dying starter? I would guess the relay. But then why does it only to that in neutral? Doesn't make any sense to me but Im sure it does for one of you hardcore mopars.