Jim is shown changing the oil on his police car, a Dodge Charger. A few scenes later, he tries to start the car and it turns over but won't start. This is attributed to a loose/stripped oil drain plug. If the car had in fact run out of oil, the motor would seize and not turn over. Additionally, the pool of oil shown under the car is too small for the amount of oil in a modern Hemi as used in the police Charger.
Jim says he left his cell phone charger on all night and it drained the car battery. The charger will only draw enough current to charge the cell phone which has a much much smaller capacity than a car battery and could not drain it to the point that the car would not start.
Longworth gets a plastic bag of white stuff out of a domestic freezer and asks whether it is dry ice. It is not possible to keep dry ice in a domestic freezer as it needs to be below -78° Celsius to be solid.
At the end, when Callie jumpstarts Jim's car, she refuses to let him do it, saying that he'll mess it up, then promptly does so herself, in two ways. She connects both cables to one battery before connecting either to the other, which can lead to a short-circuit if the other ends of both cables connect and is very unsafe, and then she allows Jim to attempt to start his car while her own is not running, which is a very good way to drain her car's battery.