Am I eligible for visa-free transit?

You need a tourist (L) visa for 1 & 2. If you are a US citizen you should get the multiple entry 10 year visa.

Thanks!
So if we didn’t do 1 & 2, do we need a visa for 3 & 4 (the cruise)?

We are all coming from the US but my group is a mix of US & UK passports.

Step 2 is causing the problem. If your route is UAE-Shanghai-Japan-Shanghai-UAE then you would qualify for 144 hours transit without visa in each direction. But if you have other stops in mainland China then you may need visas.

Also holders of British passports whose nationality is not stated as "British Citizen" will not be eligible for 144-hour transit.

Thank you for your help! I have 1 more question.

If 1&2 is an organized group tour with a company, do we still need the visa? I read somewhere that the touring company will handle this for us.

If this is true, we could fly into Shanghai use the 144 hr free transit and then go on our tour the next day, come back to Shanghai in less than 24 hrs and leave the next day on our cruise to Japan. Come back to Shanghai and use another 144 hr free transit and leave the next day to Dubai.

Is this right?

No, you still need a visa.