For most cars, the cigarette lighter is not permanently connected to the battery, but switched via the ignition key - so you won't be able to charge the battery without the key in the ignition, kind-of defeating the object.

Furthermore, given the amount of computer control on most modern cars, I wouldnt be surprised if attempting to power up the car via this switched circuit caused some unexpected and very expensive damage to the on-board systems.

Not on any of the cars I've owned - although I did hire a Ford Focus recently and that was permanently live - I know because the light on the sat nav never went out when the ignition was turned off.  Actually, come to think of it, it's a real pain, it's why my sat nav is never charged when I come to use it!

When the dashboard on my wife's Citroen went haywire many years ago (car was just a few months old - every digit and warning light came on, key in or out) - the garage claimed it was because we had been "plugging accessories into the accessory socket".  I was told that any kind of charger would disrupt the on board power management systems.