OK, heat at the battery cable says that there is a high current flowing through it, OR...there is a high resistance in it at the point of heat. Sparks indicate a high current. I guess the main point right now is that there should be zero current flowing with the key in off position. Furthermore, you say it goes away with the remote connector plug unplugged. So the question becomes what in the world is shorted to ground from that point forward and back to the motor. And if that whatever-it-is is drawing that much current, why isn't the skinny wire going to it burning up?
I am posting the complete wiring diagram. As you can see, with the key turned off, there is nothing connected to the battery. Except the alternator rectifier. Maybe that is blown? But if it is, and drawing that much current, why aren't the wires burning up? With the key turned on, there are a whole bunch of things connected (all the purple wires) BTW, the diagram confused me for a moment or two until I noticed the big plug is represented by a simple rectangular box.
I'm convinced you have at least a basic knowledge of electricity, but maybe you need some help on this?? It s very difficult for us to know what is wrong from a distance.