You're right about black Zetsu, but I don't think the white Zetsu was stated to be made by Hashirama's chakra.
Yeah you are totally right, now i remembered what I have read, I believe that Zetsu is a creation from 2 seperate entities created by Madara just so he could have an invasion force. He has been planning the Moon-Eye Plan for a while and knew he would probably have to have a military. Which does include spies and soldiers- Both of which White Zetsu is the best at it so far.