There is a heat sensor that allows the fan to stay on after the key is off to cool down the underhood temperatures. If the fan stays on as you have said, the sensor is most likely faulty or shorted.
The radiator can be flushed with water running continuously. This will flush out any loose particles. The only way to completely clean the radiator is to have is professionally cleaned in a chemical bath.
if anything is running off the battery (visibly on) the battery will eventually go dead
Disconnect the battery.
Your suspicion is correct. Sounds like the fan is running continuously. You must have that problem solved to correct the issue. The fan should stop when the ignition is turned off. Sounds like a stuck relay.
Depends if you have the ignition running or just the say you left the lights on. If you just left the ignition on all night nothing will happen except just running out of fuel. If you left your lights on without ignition running your battery will die.
Headlights run off the battery not the ignition...
Because the ignition is powered directly from the battery. The Gen/Alt only feeds the battery.
The radiator fan is designed to run continuously while the engine is running so that the temperature of the coolant can be lowered. The coolant is continuously run through the engine, collecting engine heat and dispersing it over the radiator fins; while the fan blows the fins and cooling off the coolant before recirculating back through the engine in a continuous cycle.
mainly leaving your lights on when the engine is of or leaving the keys in the ignition
It has a Generator that suppllies power to the ignition system through the Battery. New vehicles have an Alternator that makes it's own power once it is running directly to the ignition system.
check to see if the thermostat is hung open or your temperature sensor for your fan is out and causing your fan on your radiator to run continuously when key is on.
It shows that your battery is in workable condition and your alternator is charging as it should.