I would check the Cooling Fan Switch/Relay. Sounds like it is a bad switch and stays 'on' well after it is needed, if it is running the battery down.
No , for the battery to be charged the engine must be running . This means that your alternator , if it's functioning properly , is charging the car's battery as the engine is on/running .
engine cooling fan should not do this - bad temp sensor likely
If the engine is running and the alternator is good. If the engine is not running, no.
If it is running on the battery only then that means the alternator is not charging the battery.
No , just when the engine is running
Yes, and disconnecting the battery while the engine is running can severely damage the computers
By running the engine on the vehicle or by connecting the battery to a 12 volt battery charger.
Get engine running. While the engine is running disconect the POSITIVE termninal from the battery. If the engine dies, the Alterntor is bad.
Yes, because your engine is on.
use a voltmeter A fully charged 12v battery should read approx. 12.8 volts without engine running Should read approx. 13.8-14.2 volts with engine running
With a scan too or with a battery disconnect.