Julie,
You have found a baptism in the Methodist records for George for 1841.
Methodism took a lot longer to become established in Ireland as a separate denomination than in England. In Ireland there was considerable resistance to separating from the Church of Ireland. It was 1816 before Methodists agreed to conduct their own baptisms. However because of continuing loyalty and other factors, many continued to use the Church of Ireland for sacraments for many years after this date and it was 1871 before all Methodists routinely performed their own baptisms.
For marriages, the earliest ceremonies conducted by a Methodist Minister in Ireland that I am aware of, date from 1835 (Belfast Donegall Square, the first Methodist church in Ireland). However in the mid 1800s there were only a few Methodist Ministers in Ireland (Methodism relied heavily on lay preachers). So the shortage of Ministers contributed to the continuing practice of marrying in the Church of Ireland.
So to summarise, you are unlikely to find many Methodist baptisms before 1830. Few marriages before the 1840s and only a handful for many years after that. If there were no Methodist records in the location you are interested in, I would search the Church of Ireland instead, as that’s the most likely place to find the relevant event.
Aughnacloy Methodist church has baptisms for the years 1823 – 1843. (So it was one of the earliest meeting halls to keep baptisms in Ireland). If Elizabeth was born there c 1835 you might expect her baptism to be recorded in the same set of records as George. But her parent’s marriage and their baptisms are likely to be in Church of Ireland records. Errigal Keerogue Church of Ireland has marriages from 1825 onwards (copy in PRONI) but their baptisms only start in 1812 so that may be too late. Ann Liggett’s age at death points to a birth c 1809. However ages on death certificates were often just a guess so she might have been younger. Probably worth checking the early records in case she is there.
Not many Methodist Meeting Houses have graveyards and so they may be buried in public or Church of Ireland graveyards (which are open to all denominations).
The death certificate for the Ann Smith gravestone you have identified gives her place of residence and death as Ballygreenan, not Roughan.
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_returns/deaths_1897/05863/4661852.pdf
She was a farmer’s widow. The informant was her son Henry, of the same townland. This looks to be her son George living there in the 1901 census:
http://www.census.nationalarchives.ie/pages/1901/Tyrone/Augher/Ballygreenan/1725466/
There’s no Smith family there in Griffiths in 1860 but there were a couple of Morrison households, which I see was a family middle name. So perhaps the Smiths acquired a Morrison farm. The Valuation revision records on the PRONI site should tell you when the family arrived in that townland and from that you can find where the farm was, if you want.
Porbate abstra cts from the PRONI wills site:
Administration of the Estate of Robert George Smith late of Ballygreenan County Tyrone Farmer's son who died 16 May 1911 granted at Armagh to George Smith Farmer.
Probate of the Will of William Morrison Smith late of Ballygreenan County Tyrone Bank Clerk (Retired) who died 1 May 1912 at Rostrevor County Down granted at Armagh to George Smith Farmer.
The above will is on-line on the PRONI wills site. William left everything to his father George. He was single.
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_returns/deaths_1912/05368/4497023.pdf
