Chekc 72/144 hours visa free transit

I'm a French national. I would like to go to Shanghai and Beijing from Singapore, and return to Singapore. Would the following routes grant me the 72 or 144 hours free transit?

1) Singapore > Shanghai > Beijing > Hong Kong or Malaysia > Singapore (stay in Shanghai and Beijing between 3 to 6 days).
2) Singapore > Hong Kong > Beijing > Shanghai > Singapore

- for the above routes, does it matter if I go Beijing or Shanghai first?
- I would need to have separate tickets or a round trip tickets would suffice?
- if I apply for the 72 hours free transit, if I need to stay more than 3 days in Shanghai or Beijing, how do I extend my free transit to 144 hours, i.e. not more than 6 days ?
- do I apply separately two free transit at Shanghai and Beijing upon arrival at each city? or I can apply the 72/144 hours free transit when I arrive at the first city, i.e. either Shanghai or Bejing?