Sorry, but I have to (respectfully) disagree here. A 5 seconds cycling time for a compressor is way too short. Once your cabin is at the temperature you've selected you can expect a short cycling time, between 5-10 seconds, but not when you turn the a/c on for the first time. This could be due to a system that's blocked, overcharged or undercharged (which I think is your case). Only if you were unable to get hot air when selected (as well as cold air) would I blame the heater tap. cheers
ps: The A/C button will never deactivate, as it is only a request to the control module. Even with no gas at all in the system it'll turn on when you press it, but the module will not turn the compressor on (to prevent internal damage) because it will be receiving a signal from the low-press switch.