Since the govt. supplies the hashes in an unauditable way, it absolutely is controlled by the government. What's to stop them from using hashes of non CSAM material?
The hashes are supplied by NCMEC, a non-profit which is auditable, not a secret government agency.
In any case, even if a non-CSAM hash were somehow in the database, Apple reviews the images before making reports, and those reports are used in normal criminal prosecutions.
To the extent that you can say that they're not exactly a government agency, they absolutely have been deputized by the government.
Yes there is. It’s called legal liability. They are not immune to being held accountable for their actions just like any other non-profit. They may be immune from prosecution for possessing CSAM, but they don’t have any kind of immunity for damages they cause through their own actions.