The police cars used in the series are not often of the type used in the Los Angeles Police Department during the 1970s.
Occasionally a car may run off a cliff or crash and then explode. Gasoline is volatile but not explosive, so there is no reason for the cars to explode.
It's never explained why Jedediah Romano "J.R." Jones does not carry a gun.
He obviously knows how to handle a firearm as he does sometimes after taking one away from a bad guy. A law abiding citizen over the age of 21 as well as a military veteran working for a licensed private detective agency should certainly qualify for a gun permit.
Barnaby is renowned for his scientific ability. However he has a habit demonstrated several times of blowing into the evidence bags to open them up. This would contaminate anything he put in the bag.