Beijing-Shanghai- Seoul-Tokyo-Taipei-Hong Kong

My husband and I are traveling into Beijing, planning 48 hours there, then flying to Shanghai for 24 hours, then out of PRC to Seoul, Tokyo, and Taipei. We have a long layover in Hong Kong after Taipei and are planning some sightseeing there before we fly back to the US via Beijing. On our way back we will not leave the airport as our layover is only 2 hours. I am not sure if we will need to apply for a visa or not- we will be in the mainland for less than 72 hours but are planning two cities- will we apply for the 72 hour free transit at each airport (Beijing and Shanghai) or is that not allowed? From what I've read we will not need one for HK. We would really appreciate any insight! I will be traveling with a US passport, and my husband on a Brazilian passport, if that makes any difference. Thanks in advance!