No, Puddles of Mud is not a Christian band. Some of their songs may have a Christina meaning, but the band itself is not Christian.
I know they bear a striking resemblance but edge is not the lead singer of puddle of Mudd. His name is Wes Scantlin.
Can't find an exact date as to when Flanders and Swann wrote it but they were active between 1956-1967. (I believe it was recorded in 1959) see the related link below for more information.
10 Years, Sevendust, and Chevelle.