Hello

Part one:

Jeff, you are definitely allowed to travel your planned itinerary:

US - Hangzhou - Taipei

Just to clarify one detail. You are allowed to use several airport within Shanghai transit area. In other words, you don't have to pay more for a direct flight to Hangzhou because you can use Shanghai too. In reality, the following airports within Shanghai transit area are allowed for any kind of combinations: two Shanghai airports (Pudong and Hongqiao), Nanjing, Hangzhou and Ningbo. You are allowed to arrive on any one of them and also to depart from any one. In other words, you can arrive at Pudong airport and later depart from Hangzhou. Just take a look about the available tickets and their prices and just be relaxed.

Also, you don't have to be worried about Taipei as your following destination because you can travel there perfectly legally and also it is considered, within 144 hour visa free transit rule, undoubtedly eligible and without any negative connotations. In other words, you will have no troubles leaving mainland China to Taiwan.

The only thing you are not allowed to do is to have any additional stop within mainland China beside the airports within Shanghai transit zone, of course. In other words, you cannot fly:

Seattle - Beijing - Hangzhou - Taipei

Part two:

Of course, you can fly:

US - Nanjing or Shanghai or Hangzhou - Taipei

Also, you should keep in mind that 72 hour visa free transit has been replaced in Shanghai area by 144 hour visa free transit a couple of years ago.

You will print out your flight tickets and bring those printouts with you. Also, you will declare at check-in in US that you will travel to Taipei via China using 144 hour visa free transit. After a short inspection of your tickets, you will be allowed to board the plane.

Upon arrival in Shanghai or Hangzhou, you will approach 144 hour visa free transit counter where you will fill out a blue form where you will write down the identity of your friend if you intend to stay at her/his home in Hangzhou. Also, you will also declare an address and her/his phone number.

Please, keep in mind that you will have to register within 24 hours at the relevant police station in Hangzhou accompanied with your friend. Basically, this is it.