Do we need to apply for any permit or notify the airline during check-in to go out of the airport?

Evangeline, you are qualified for 72 hours free transit in Beijing. You needn't notify the airline in advance. After landing, you just need to following the signs to find the window where you can apply for 72 hours free transit. Just show your onward air ticket and passport and then you will get a stamp on your passport. With this stamp, you can leave the airport to stay in the city.