Yes, it is possible to charge the battery through a cigarette lighter socket, but there is a couple of caveats:
The car's circuitry must have the lighter socket "live". It may require setting the ignition switch to "accessories."
The charge rate must be kept low enough not to blow a fuse associated with the lighter socket (often 10 Amps). At 10 A it is going to take several minutes to get a depleted battery charged enough to crank the engine successfully.
There are commercial products that work on this principle.
However, it seems you do not so much want to charge the battery but to keep it charged while running electrical devices. This is possible, but the 10 A limit means you can not go over 135 W of sustained total power draw without depleting your battery.
The air conditioning compressor is driven by a belt drive from the engine, so you cannot run that from a purely electric source. There may be AC compressors that are powered by AC electricity or by the engine.
The blower that circulated the air is electric and can be powered from the battery. The dashboard is mostly electric, the only bits I can imagine being of use in a stationary vehicle are the entertainment system and the clock. They can be powered this way.