Starter keeps starting!...

You need to do some electrical testing on your ignition system and starter system. I'd start by disconnecting the wires from the starter and attach a voltmeter to the solenoid wire and see if it still does the same thing (as in, still has power at the solenoid wire after turning the ignition switch to the start position and then letting it return to the run position). If it still has power in the run position, you'll need to trace back where it's getting it's juice. As Dana67Dart mentioned, you may have an ignition system wiring problem that is contributing to the starter system problem. Start by isolating the two for testing purposes. Tracing why you don't have juice at the coil in the run position may solve the problem too. You may have the coil run electricity hooked to the solenoid somehow.