What does "to leave the car out of gear" mean? Is it the same as "to disengage"?
And, is "disengage" used when the gear is not engaged? (Any informal term instead of 'disengage' please?)
Yes, disengaging the gears means leaving a machine out of gear. The commonest way of saying this is "Put/leave the car in neutral".
Note that you can also disengage the brakes, the clutch etc. so 'disengage' by itself doesn't necessarily mean the gears.