I have had a quick look at the information you have given us and assuming the later births are all correct, it seems ok up to "Henry Whitaker Green FLETCHER". Without going in to too much detail -
Isaac Green FLETCHER (birth registered as Isaac GREEN), mmn BAKER 1871. He can be found in various census with wife Hannah (WESTWOOD) or parents Henry and Dinah FLETCHER.
Henry FLETCHER married Dinah BAKER in 1861, registered in Dudley. They can be found in various census, he is an iron puddler, born Dudley c1833. He seems to be with parents William and Charlotte FLETCHER in 1841 and 1851, though according to ages on the census Charlotte must have been around 15 when she had Henry (though her age does vary across the different censuses, as does William's). William is an engineer, at the ironworks.
William FLETCHER married Charlotte GREEN in 1833 in Tipton. I can't see a baptism for son Henry, though there is a Henry Whitaker GREEN baptised at Kingswinford on 15 Aug 1832 to mother Charlotte GREEN (no father mentioned). I haven't looked in detail but I'm not convinced that this is "your" Henry, and if he is, from the names it doesn't sound like William FLETCHER is necessarily the father of Henry Whitaker GREEN.
I'm also not sure about the death you have for William FLETCHER in 1883. I can't see him after the 1861 census, and there are a number of possible deaths in the Dudley registration district that could be his.
Does this agree with your own research up to this point?
