Most of the above seems accurate except for the CAAC Background check. Having been here in China 18+ years myself I can tell you that this particular check was much more involved that what is stated.
For myself and others I know the FBI check was not required nor accepted. What was insisted on was a Criminal Records Check (CRC) Specifically from one's Home Town. As in the Town you were born in/grew up in.
This CRC once acquired must then be Notarized.
Then it must go to the Dept. of State for authentication to verify that the Notary public is actually authorized.
Lastly, it must then go to the Chinese Consulate or Embassy Office for the region your hometown is located in to then be Authenticated again by them.
Only after all this is is your CRC ready for submission