County lord manor, or Duke manor are correct since:
Emperor -> Emperors Family (Royal Family) -> Kings -> Royal family -> Dukes -> Other nobility (depending on system) -> Commoners -> Slaves (if exists in the system) OR
Emperor -> Royal Family -> Dukes -> Other nobility (depending on system) -> Commoners -> Slaves (if exists in the system) OR
Kings -> Royal family -> Dukes -> Other nobility (depending on system) -> Commoners -> Slaves (if exists in the system).
So by the way it often works so are Dukes some way or another connected to the king or emperor by family.
Children of the world, or Crown prince works since:
If it is an emperor or king so is it the "top" of the world, so children of the world works from every successfull decendent direct or indirect, the way crown prince can get used is either for the emperor, king or Duke/similar high ranking manors/families as the successor.