Viv,
Jane Graham’s address on the marriage certificate is Lisnisky, parish of Seagoe. So the 2 deaths I have found for William & Rachel are almost certainly her parents. So if you want to find Jane’s baptism, and that of any siblings, you’ll need to get someone to look up the Presbyterian baptism records in PRONI. (There are plenty of researchers who will do that, for a fee).
I had another look at Lisnisky today on Google Earth. The land where the Graham house was is still agricultural land but there are no houses on it now at all. All have been demolished or collapsed. I can see some trees and open farmland but nothing else. To get to it today you would need to walk through a couple of fields along a farm track up from the nearest farm.
Joseph Hawthorn was living in Tamnaficarbet townland, parish of Seagoe. The modern Carbet Rd runs through the middle of it. That’s about half a mile north of Lisnisky. Most of that area is an industrial estate these days. (In the 1960s they built a dreadful new town called Craigavon in that area. It’s all light industry, roundabouts and soulless residential estates.) I had a look at the Valuation Revision records for that townland around 1876 but I don’t see the Hawthorn family listed. Most of the townland was farmland with some weavers’ cottages. Most of the tenants’ names changed over the period of the record, indicating a high turnover of occupants. The clerks who compiled these records only visited every couple of years so if you weren’t there long you might be omitted altogether. Or of course, Joseph might just have been lodging with someone. In which case he wouldn’t be listed at all.
https://www.nidirect.gov.uk/services/searching-valuation-revision-books
There were 29 houses in Tamnaficarbet in the 1901 census, containing a total of 96 people. Most were labourers or weavers:
http://www.census.nationalarchives.ie/pages/1901/Armagh/Kernan/Tamnaficarbet/
There’s no sign of James Hawthorn in the area in the 1901 census, so I suspect he died before that. I see a death for a James Hawthorn on 20.1.1892 aged 79, registered in Lurgan, which seems about right. There was also a death of a James Hawthorne on 27.3.1889 aged 50. (Too young I think). The 1892 death looks the best but it isn’t on line free. So you’ll need to pay to view it. You can view the original certificate on-line on the GRONI website, using the “search registrations” option:
https://geni.nidirect.gov.uk
You will need to open an account and buy some credits. It costs £2.50 (sterling) to a view a certificate. If you decide to look at the 1892 death, let me know the details. It will tell you whether his wife was dead or alive, and so which side of 1892 to search for her death. The informant’s details may be helpful too as that might be his wife, a son or a daughter.
I notice from the most recent information that you have given above that Joseph was 83 when he died, so born about 1844. And that he was born in Co Down. So not originally from the Lurgan area. So maybe his parents didn’t live in the Lurgan area. If Joseph was born c 1844 then his parents almost certainly married before statutory marriage registration started in 1845, and so their marriage may be difficult to find. I assume you have already looked for it on rootsireland. They don’t have all the possible records though. Many churches records have yet to be put on-line. But if you don’t know which church they married in, it’s a needle in a haystack finding the right one in PRONI. Co. Down has hundreds of churches.
Anyway I hope this helps a bit.