I am unsure about this family in Kilsyth. First of all, the children were born in Stirlingshire, not Linlithgowshire (West Lothian) and James' occupation of schoolmaster and registrar doesn't sit comfortably with being a coal miner.
James H and Bethia Kennedy also had a daughter Helen Mitchell Henderson, born 19 March 1864 in Kilsyth. Bethia and Janet are in Kilsyth in 1871, but James is not. I think he must be the one who died in 1866 in Kilsyth. Noting that he had a daughter Helen Mitchell and that he was born in Airdrie, he has to be the son of William Henderson and Helen Mitchell, baptised 10 September 1809 in New Monkland. But I have yet to be convinced that his son William is your William Henderson.
I looked for Agnes Marshall, mother Barbara Smith, and I found Agnes, daughter of John Marshall and Barbara Smith, baptised in Kilysth on 25 December 1824. That taillies exactly with the wife of James Henderson, who was 26 in 1851, apart from her birthplace which was transcribed as West Lothian, Linlithgow. Is that some sort of error, or was she born in Linlithgow but taken back to the family parish for baptism?
I had a look for an illegitimate John Casey in 1870/1874, mother Margaret, but didn't find a likely candidate that way.