You should consider a different set of flights because it does not look like your itinerary is eligible for the 72 hour visa free transit.
I cannot find official Chinese guidelines but multiple sources on the Internet allow us to form a summary. Sources - Travel China Guide, Travel China Cheaper,and a blog post. The consensus of all these is that the following must hold for a visitor to use 72 hours visa free transit (emphasis mine)
- You must be national of one of the 53 qualified countries. US, Schengen, UK, Russia, Canada, Brazil, Mexico, Australia, New Zealand are all included in this list.
- You must be transiting in one of the following cities - Beijing, Chengdu, Chongqing, Dalian, Guangzhou, Guilin, Hangzhou, Kunming, Nanjing, Shanghai, Shenyang, Tianjin, Wuhan, Xiamen, Xian
- You cannot leave city limits during this period. There are some exceptions to this rule - E.g. passengers transiting in Guangzhou, Chengdu, Qingdao, or Changsha are allowed to travel in the whole province.
- Your itinerary must consist of flights from Country A to the designated Chinese city and onwards to a separate country/region C (within 72 hours). For this purpose Hongkong, Macau and Taiwan are all considered third regions so your flights are allowed to go there.
- You can only use flights for entering and exiting China.
- Inbound and outbound flights must not transit through any other Chinese airports.
The last point is emphasised by the news report about two kiwi citizens who had a harrowing time beacause their inbound flight had a (unannounced?) stop-over in another Chinese city.
To answer your specific questions:
- How are same-plane, same-flight-number flights like MF857 counted?
There is an advisory article from a US immigration law firm which states explicitly
Your departing transportation must not make a further stop in China,
even if you are not required to change planes. For example, Beijing ->
Kunming -> Hanoi doesn’t qualify.
so I would say such a flight will not be counted as valid.
- Does China do exit controls on such flights?
I think the broader issue here is if you will be issued the visa-waiver stamp when you land in China? If you present this itinerary and officials determine this is not valid (as outbound plane has a halt in another city), it is very likely they will refuse you entry and deport you back to Taipei.
To be safe, take a direct flight out of Xiamen to a third region (Japan?) and then onto LAX if you can't find a direct flight to LAX.