Multiple visits to American citizens living in Shanghai

Hi, you can apply for dependent Z visa accompanying your daughter's Z work visa. It is possible to get 2 year multiple entries, but the final result depends on consular officers. Documents required are invitation letter from her employer, and proof of relationship if you apply at the same time with your daughter. If you apply later when your daughter entered China, the copies of her Z, work permit and residence permit in China are required.
Plane reservation for each visits are not required in this situation, but it will be better if you can provide any. With US passport, you are not required visa or permit to HK.