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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.
The starter will not stop the engine from running once it is started. You have another problem.
With the flywheel spinning at about 1,000 RPM at idle, the starter, if engaged with the flywheel, would be forced to spin between 15,000 and 20,000 RPM. Once the engine has turned over and is running, the overrun clutch will release the starter from the flywheel and prevent the gears from re-meshing (as in an accidental turning of the ignition key) while the engine is running
If clicking while trying to start then might be a bad starter. If clicking while running, then might be valve/lifter noise.
Never disconnect a battery while an engine is running!
At the back end of the engine is a large ring gear connected to the engine crankshaft. If there is a starter (some engines do not have starters) it will be mounted in the engine such that the bendix gear of the starter can engage the ring gear and spin the crankshaft. It's called a bendix because it jumps out to engage the ring gear (while the starter motor is running) and retracts off of the ring gear once the engine is running (starter motor no longer running). It is typically access from under the car. If you must replace it, let the engine cool off first. Otherwise the starter will be just as hot as the engine is.
No, I would seek another answer to the problem. If the starter starts while the car is running, it may grind the bendix or the starter into a toothless nub and the starter will quit functioning alltogether.
Yes. The alternator is used to keep the battery charged, while using the headlights, radio, a/c, etc. The starter to me is misstated because it actually cranks the engine. The ignition system starts it, and keeps it running.
Bad ignition switch or starter solenoid
The remote just connects the starter selenoid with the battery, when it completes the circuit, the engine starts and will run until it runs out of gas or the engine is disabled.
If the engine is running and the alternator is good. If the engine is not running, no.
The compressor does not run all the time while the engine is running. The compressor has a clutch which is activated on demand. The pulley may be spinning but that doesn't mean the pump is engaged.