Do churches of Christ register with the government as a whole or each congregation independently?

Answer:

The United States government does not require churches to register with it. The closest it comes to this is that it offers an exemption from taxes to non-profit organizations, which would include churches. Each congregation independently chooses whether to register for that exemption or not, but no church is forced to register or even needs to register. See Are you a 501C church?

If we were in a country that requires registration, then each congregation would follow the government's requirements, so long as those requirements did not interfere with the church's business. When government does try to restrict a congregation, we simply ignore the government in whatever matter they are wrong in regards to the church. We do so pleasantly, calmly, and quietly.

There isn't much interaction between churches. If members of a church come to this one teaching a false doctrine, we will contact that other church to find out if that is what they are actually teaching or not. If they are teaching falsely, we will try to persuade them to return to the truth. If they refuse, we simply cease to interact with them. See Church Autonomy and Cross Congregational Fellowship.