The system is pre-programmed (which works best for me btw), as long as you keep it on auto-modus.
Once you touch one of the controls, the system is on manual. Then you override the program. Except when it auto-detects a misty windscreen, if on-board, or a too long period of the re-circulation flap closed, than it will control that part again.
The FULL indication was there before to show everything was controlled: temp, fan speed, etc. But once you overrode by pressing one of the buttons, like air flow direction, the full indicator would go off.
The A/C compressor can only work by a running engine, it's belt driven. Upon the pressure inside the closed system, up to 15 bars, it will switch on and off to maintain that pressure steady. This switching on and off is automatically by a sensor in the piping (front right, below side of your condensor). This is sensor activates the clutch of the A/C wheel, driven by the auxiliary belt (multi-V).
Hence that once you switch off ignition, your climate system cannot run, including A/C part of it.