The generator in a car doesn't run for free. It is powered by the engine and it's operation while charging is felt as a braking effect on the main engine. This additional drag is produced by a phenomena known as back electromagnetic force (back emf) that causes the generator to resist turning while generating current to power electrical devices in the car or recharge batteries.
Another way to realize that using electricity in a car isn't free is to consider why it wouldn't work to add a number of these generators and then use them to power electrical motors to propel the car. There is no free lunch.
On the other hand (and not really related to the points I'm making above), stopping a moving vehicle reduces the overall energy of the system because the kinetic energy is proportional to the square of the vehicle's velocity. Due to conservation of energy this energy has to go somewhere. Conventionally, this stopping energy is dissipated through heat (and to some extent breaking down a bit of the the brake pads). Some vehicles, like a Tesla, are able to capture a portion of this kinetic energy during braking, but its complicated, see [1].
[1] https://www.tesla.com/blog/magic-tesla-roadster-regenerative...
No. Your use of energy from the generator through the cigarette socket manifests as a braking force in the generator. The more energy you use, the more it brakes. The car's engine has to work harder to overcome that braking.
The vast majority of cars already have a small electrical generator installed. Plugging things into a cigarette socket to access that electricity causes 0 extra load on the engine.