Let me put your mid at ease...
If you spin a generator that is not hooked to anything, all you are wasting is the energy needed to overcome the bearing friction of the generator. If you then attach a load to the generator, you'll notice your engine spinning the generator will slow. That is the electrical load causing the generator to be harder to turn, slowing down the engine turning it. In other words, as you absorb energy from the generator terminals, the engine has to work harder to turn the generator.
So, if you have an engine idling, and you jump to a dead battery, all of a sudden there will be more load on the generator, which will slow down the idling engine. That engine's computer may compensate, by opening the throttle, thus increasing gasoline intake.
The upshot: if the car and generator are working properly, then you pay in gasoline for all the energy you use. There may be other losses in the system, but they will only increase the amount you pay, never decrease it. All you have "free" is higher electricity production capacity, and actually youo pay for that, too, in higher parts costs and more friction/heating losses (i.e., lower efficiency).
There is no such thing as free electricity in a car's engine-generator system.