Are we okay to apply 72 hour visa July 5: US > Shanghai > Japan & July 16: Japan > Bejing > US

Since you are a US citizen, you're good to go up to 144 hours, per the info on this website, pasted below:
"144-Hour Free Transit in China
China's 144-hour free transit policy is carried out in Shanghai, Jiangsu, Zhejiang, Liaoning as well as Beijing, Tianjin and Hebei, to facilitate international travelers. It allows passengers from 53 countries or regions to transit in either Shanghai-Jiangsu-Zhejiang region, Beijing-Tianjin-Hebei region or Liaoning Province for no more than 144 hours (6 days) without holding a visa."
Let's wait for other folks to confirm!!!

You can apply the 144 hour free transit for both layover in Shanghai & Beijing. Nice trip.