No no! You cannot combine multiple transits without visa on one visit to China. You must qualify for a single type of TWOV in order to be allowed to board the plane from your point of origin.

If yo have multiple stops in China, the ONLY TWOV applicable to you is the 24-hour TWOV (with entry permit). This allows multiple stops and transfers, including domestic flights. There is no such thing - that I have yet to see - that qualifies as a true 'technical stop' in China (despite what airlines might call it). From other flights, and everything I have read about the Xiamen air stops in TAO, at this stopover point (TAO) you will enter China and pass through immigration. (In this respect it is the same as the USA... first point you touch US soil you need to be cleared.)

As you have a stop in TAO, if you do not intend to buy a visa, you must depart China no later than 24 hours after arrival. This is actually the time your plane is scheduled to take off... not the time you arrive at the airport or clear immigration.

As you will enter China in TAO under the 24-hour TWOV, you cannot then apply for the 72-hour version in XMN.

Your options are (a) apply for a visa (b) change your itinerary to stay in China less than 24 hours or (c) change your itinerary so that you only have one stop and can avail yourself of the 72-hour TWOV.

moondog is correct - if you are going to apply for a visa you would just apply for a regular single, double, or - in the case of US passport holders - a 10-year tourist visa. I'm not even sure of why people would apply for a transit (G) visa.

24-hour TWOV is available to almost all nationalities. 72 and 144 hour TWOV are limited to 53 nationalities.

You are correct that you must register with police within 24 hours of arrival, but if you are staying at a hotel they will do this for you. If you are staying with friends they will need to accompany you to the police station to register you (they will need to bring personal documents).