Do 2 roundtrip airfare qualify for 144 Hour Visa-Free entry? Need to figure out how to book tickets.

Anna, your presentation is a bit confused, but if I have understood you properly, you are asking if you can do the following itinerary:

New York - Shanghai - Astana - Shanghai - New York

Or in other words, you plant to have roundtrip Shanghai - Astana - Shanghai within the roundtrip New York - Shanghai - New York.

If my understanding is correct and based on the dates you have presented, I can just confirm that you are definitely eligible for 144 hour visa free transit in Shanghai twice or better to say that the first time in Shanghai you will be eligible for 144 hour visa free transit and during your second arrival in Shanghai you will be covered by 24 hour visa free transit since your layover (transit time) in Shanghai will be shorter than 24 hours. This is fine, eligible and no visa is needed.

Just one important, additional information. During your second arrival in Shanghai, you will have to exit the airside because since you are flying in separate tickets, you will have to exit the airside in order to collect your luggage and to perform new check-in for your flight to New York. This is all fine and the first thing you will do upon your landing from Astana is to approach the transit counter where you will apply for a temporary entry permit in order to be allowed to exit the airside to perform new check-in and all other formalities. Here you will present your flight tickets and after a short check, you will be granted a free temporary entry permit.

In the end, I would like to emphasize one detail... There are no direct flights between Shanghai and Astana and vice versa, so you should be careful that your itinerary Shanghai to Astana and vice versa does not have any additional landing within mainland China because if you plan something like this, you would be ineligible for 144 hour visa free transit!