questions from Sal, an UK citizen travelling Beijing for under 72 hours

hello, i am an UK passport holder travelling to Beijing as a tourist;
i am flying into Beijing from London on a wednesday morning about 5.30am,
then flying out of the same Beijing airport 2 days later on the friday about 12 noon to Seoul, South Korea;
i have my tickets booked for Seoul and I do not need a visa for South Korea.
i am staying in a hotel during my 56 hours or so stay in Beijing.

i believe i qualify for the 72 hrs free facility?
if so, do i just apply for, and obtain, the free transit at Beijing airport ON ARRIVAL or is there any applications/enquiries with embassies/ my airlines BEFORE my arrival?
is there a charge for this free transit and if so how much?
also, i want to travel out of the IMMEDIATE Beijing area to see the Great Wall, is this ok to do?
also, do i need to register at a local police station or is my hotel automatically going to register for me?
please advise me with my above queries, many thanks.