I'm wondering how laptops turn themselves off after shutdown.I don't understand you question. ATX is a standard for PCs and PC power supplies. No relationship to laptops that I'm aware of.
You mean a power supply built into the motherboard?There are hardware registers that are controlled by software that interface to the power supply to control that.
For a laptop, the "power supply" is external. The little brick thing.You mean a power supply built into the motherboard?