We don't have any information on your vehicle so are not able to tell you how hard it is to remove the front bumper. Generally front bumpers are not too difficult to remove if you know where to look for clips, bolts and nuts holding them in place.
As to the fan working when ignition is off, if the fan works and goes off after a while, that is normal. If it stays for a longer period and goes off, it could mean an overheating issue or you don't have sufficient coolant in system.
If it goes off with vibrations from closing the bonnet lid, yes, it could be a sticking relay.