To what degree were Japanese religious beliefs challenged and/or changed by the introduction of different religious and philosophical ideas from China?
Before Buddhism arrived from China, Japanese followed the Shinto faith exclusively. When Buddhism arrived, it became very popular but did so without really destroying Shinto. Instead, Shinto and Buddhism became combined in many ways.
Buddhism was able to accomodate most Shinto beliefs. For example, the Japanese had worshipped the sun goddess, Amaterasu. When Buddhism started to gain strength, Amaterasu simply came to be identified with the Buddha Vairocana. Because of this sort of synthesis, the two faiths were able to coexist.
The spread of Buddhism is given credit for changing some aspects of Japanese culture. Japanese came more and more to cremate their dead instead of burying them in tombs, for example. Buddhism is also said to be the cause of the decline in Japanese eating of meat.
So, the coming of Buddhism from China altered some aspects of Japanese religious and ethical life. But it absorbed previous Shinto beliefs rather than supplanting them.