I found a likely marriage for James Green and Elizabeth McMullan (Mary Elizabeth’s parents). It was on 30th March 1854 in Bovevagh Church of Ireland (which is an episcopal church), between Limavady and Dungiven in Co Londonderry. Tradition was to marry in the bride’s church, so that may be where Elizabeth was baptised. James was a labourer living in Burnfoot, and Elizabeth was living in Ardinarive.
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_returns/marriages_1854/09477/5434657.pdf
There was another couple of exactly the same names married in Cork in 1852. However he was an engineer, not a labourer, and so I very much doubt it’s your family:
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_returns/marriages_1852/09428/5415295.pdf
Looking at Griffiths Valuation for 1858 I don’t see a Frederick McMullan in Ardinarive but there was a Catherine McMullan who was probably a widow. That might be Frederick’s widow. She had a labourer’s cottage on plot 2d which today is just all moorland but which in the 1800s had some labourer’s cottage son it. It’s up a dead end lane off the Polly’s Brae Rd.
In 1879, James Green was alive and was an iron foundry labourer. There are no iron foundries anywhere near Londonderry and so my guess is he was working in Scotland (where there were hundreds of foundries).
There are several trees on Ancestry with this Green(e) family. One, (the Green-Houston family) has James Alexander Green lodging in Govan in Glasgow in the 1871 census. It also records that he had a sister Martha 1835 – 1896 (died 13.9.1896 Belfast) and a brother William 1830 – 1919 died Limavady. The tree does not record James’ wife’s name or any of their children. James parents are listed as Hugh Greene and Mary Fowler (no dates).
Another tree (Greene family) does have Elizabeth McMullen listed as James’ wife together with the 1854 marriage I have found. However there is no further information on James or Elizabeth after 1854 on that tree.
There are several trees which have James Green but which are private so they might have more information. I can’t tell. You can e-mail the owners and ask for information though.
Regarding Scottish background, given that the Green & McMullan families were protestant and lived in Co Londonderry, the chances are that they originated in Scotland and came to Ireland in the 1600s (as did 200,000 Scots). They have been going back and forth to Scotland ever since and still do to this day.
I don’t have a Skype account but I do use Zoom. If you e-mail me on ahoghill@irelandxo.com we can probably set something up that way.