Nothing happens if you re-pull the trigger low during the 5 minute window. The output will be off after 5 minutes regardless what you do with the trigger input - you can even keep the trigger low constantly and the output will go off when the time is up anyways.
Absolutely NOT. The Falstad simulator is wrong.
The datasheet says, "The trigger input must be returned high at least 10?s before the end of the timing interval." Or it will never time out.
An applications note from Philips (they bought Signetics who invented the 555) shows a coupling capacitor used to feed a negative-going pulse to the trigger pin and a pullup resistor: