Not true. There are numerous ways a LEO can legally search your car without your consent or a warrant. Granted, it doesn't happen often to average Joe who isn't involved in illegal activity but it does happen.
CA statutes specifically state that glove compartments or utility compartments DO NOT qualify as suitable storage locations.
Yeah, there are laws. Statutes also say it's OK to remove the gun from the car as long as it remains in the locked container.
(Penal Code ยง 12026.1.)
1. the firearm is within a motor vehicle and it is locked in the vehicle's trunk or in a locked container in the vehicle other than the utility or glove compartment,
2. and when the firearm is carried by the person to or from any motor vehicle for any lawful purpose, the firearm must be contained in a locked container while being physically carried.