Entry into china visas requirement while cruising with US Passport

Hi,

We planning to fly into Hong Kong from the US to embark a cruise ship with an itinerary starting from Hong Kong with port of calls in Taipie, Taiwan-->in Nagasaki, Japan-->in Busan, Cheju, Seoul South Korea-->Beijing, China (dock for 3-days)-->Shanghai, China. Shanghai is where we disembark from the cruise and go directly to catch a flight to Tokyo, Japan. Which China visa exemptions are we qualified for based on our itinerary? Is does not appear that we can use the 15-day free transit since our first entry from the cruise is Beijing.

Do we need visa to enter Hong Kong to embark the cruise ship, another one to disembark the cruise for the 3 days the cruise ship dock in Beijing?