If you are sure that all the wiring is ok, then check the fuel.
Never!!!
Your starter solenoid is sticking. Replace the starter solenoid and you should be good to go.
NO! It should not be running.
the starter should not stay engaged after the vehicle is running. more than likely the armuertor or the bendix is faulty. replace the starter.
If the starter is whining while the engine is running at an idle you have a problem with the ignition switch or the starter solenoid is stuck. The starter should not be engaged while the engine is running unless you are holding the key over on the start position.
It should be on the transmission on the passenger side of the vehicle. It is easy to find because there should be wires running from the alternator to the starter solenoid.
Yes, absolutely, always give the vehicle ahead of you room to maneuver especially a motorcycle. Your car or truck has no problem running over a motorcycle and its riders.
If the starter keeps running when the key is removed, you have two wires touching that should not be. It can also be a bad solenoid. Either one can start a fire.
The alternator is irrelevant to the starter.... if you have an electric starter, it's fed from the battery directly... which kinda makes sense, seeing as the alternator only produces power when the engine is running. Get a digital multimeter... set it to measure voltage. Put the black contact on a good ground point, and put the red contact to the stud on the back of the alternator. You should see 13.8 to 14.3 volts.
A small flywheel needs to stay with the starter that goes to it. You can use either flywheel, and the block should be drilled for either starter.
The 1991 Ford Explorer should have a separate starter relay that sends power to the starter. It sounds like the relay has stuck and is sending power to the starter even after the vehicle is running.
you really should just to prevent having a flat tire in the future. The tubes can also dry out to its breaking point