Audrey,
I hope I did not give the impression that I knew all that info.
I got it all from the Internet. We are so lucky to have it.
By the way it is usually a frustrating task to find out where in Ireland, our folk came from.
Here's a laugh... my oral history was that my Gallaghers were from Co Mayo. I paid two visits there before I discovered from records that they actually came from Co Sligo.
One tactic is to track all family mambers through all the censuses, hoping that at least one emunerator slipped in a county as well as "Ireland".
I think William's brother, James married Sarah Shaw, born Haltwhistle.
Catherine's death was registered in/at Hexham DRO in December Quarter 1879, aged 53.
On 1891 census in Tanfield (4087-50-1) there is a widowed William Rafferty, 62, lodger, colliery labourer, born Ireland.
No other good candidates for your William.

Good Luck
Michael Dixon