William's birthplace of Heaton & Helton is most likely to be Hetton which was a small area near Kelloe.
DCC has an online BMD index (not complete) but I can't get it to load at the moment.
http://www.durham.gov.uk/Pages/Servi...?ServiceId=663
There's 2 baptisms of Cooper children with parents Peter & Frances Cooper on Durhamrecordsonline, John (1841) & Isabella (1846).
http://www.durhamrecordsonline.com/index.php