Flight from USA to SH to Beijing to USA

Jenny, the most important detail in your itinerary is that it is not eligible for 240 hour visa free transit and if you take this trip, you will have to obtain a Chinese visa in advance. As you can see, your itinerary is not a transit type because you miss the 3rd country. Your trip is a round trip based on a return ticket. In order to be eligible for 240 hour visa free transit, you will have to use an itinerary that is eligible, for example: US - Shanghai - Hong Kong - US. In other words, immediate points before and after mainland China have to be in two different countries or eligible territories (eligible territories are Hong Kong, Taiwan and Macau). As far as your stay at your daughter is concerned, this is allowed and is done by many. Upon your landing in China, you will present that you will stay at your daughter and you will supply the authorities with the address, name and the phone of your daughter. You will go to your daughter and within 24 hours of your arrival, you will both go to the closest PSB (Public security bureau) where you will make a registration. After this, you do not need to perform any additional formality. Now, it is important to make an itinerary that will be eligible or to obtain a Chinese visa. If you create an eligible itinerary, you are allowed to travel across the provincial borders without any problem or limitations, so you can freely travel to Beijing without any formality. Hope you will manage to create an itinerary that will allow you to visit your daughter visa free.