pinkie,
Statutory birth, death and marriage registration (in some jurisdictions called Vital Records) only started in Ireland in 1864, save for non RC marriages which were recorded from 1845 onwards. So you probably won’t find many statutory birth, death or marriage certificates in Ireland for this family. For earlier years you usually need to rely on church records, where they exist. Not all churches have records for that period and not all are on-line.
There are copies of most surviving records for Fermanagh in PRONI, the public record office, in Belfast. A personal visit is required to access them. Access is free. This link explains what records exist, parish by parish:
https://www.nidirect.gov.uk/publications/proni-guide-church-records
If you are unable to go yourself, you could employ a researcher. Researchers in the PRONI area: http://sgni.net
Your family were Church of England (which in Ireland is the Church of Ireland). You say they lived in Magheraculmoney parish. That’s an area around the village of Kesh on the shores of Lough Erne. That parish has baptism, marriage and burial records from 1767 onwards. I don’t think they are on-line anywhere but there is a copy in PRONI. If you can get someone to search those, you may be able to confirm the baptisms, marriages and find burials of the various Armstrongs you mention. The records may also give you a townland (address) which would help establish where they were living then. If any married post 1845 or died post 1864 you might also get statutory certificates but you would need to have some idea of where they were living to be sure of finding the right person. Magheraculmoney is in the civil registration district of Irvinestown. I searched for Armstrong deaths there 1864 – 1901. There were 235. 932 Armstrongs in Co. Fermanagh in the 1901 census.
There are few other records for Fermanagh in the 1700s (and practically nil for the 1600s) so you won’t find a lot of additional information, I fear.
You mention one Armstrong possibly going to the US and another to Scotland. There are unlikely to be any records in Ireland confirming their departure and so you would probably need to hope that someone has them on a tree on a public family to locate them. Sarah Armstrong (1792 to 1886) is on 12 trees on Ancestry. One (the Stead tree) has her father Robert (born 1748) dying 3.12.1817 in Drumcaghy (so you might want to search for a burial record for him to confirm that). It also indicates that some of the children are half siblings suggesting Robert Armstrong married twice. Thomas Armstrong (b 1790) is shown as dying 17.1.1845 in Ohio, US. Sarah is shown as being born in Tiriwinnee but I am unable to find anywhere of that name. I think it may have been mistranscribed. Drumcahy does exist though. It’s on the modern Edenclaw Road a mile east of Ederney Village. In 1901 there were 11 houses there, population 36. No Armstrongs.
http://www.census.nationalarchives.ie/pages/1901/Fermanagh/Magheraculmoney/Drumcahy/
Co Fermanagh and some adjacent counties was very heavily settled in the early 1600s by people “encouraged” to move there from Scotland and Northern England.
When King James I became King of England in 1603, he was already King of Scotland and so then became the first King of both countries. For hundreds of years the Scottish Borders had been fairly lawless and travellers were routinely robbed, and cattle often stolen and herded across the border by moonlight. The folk from that area were known as Border Reivers (reiver is a Scots word meaning a robber). The new King was particularly keen to stamp this out because he saw it as an obstacle to commerce between the 2 countries, and being joint ruler that bothered him more than his predecessors.
At the same time he had the problem of Ireland. The Spanish Armada had recently attempted to invade England and further invasions by the Spanish or French were feared. Ireland was seen as a possible jumping off point for such an invasion and understandably, the native Irish could not be relied on to support the English or resist any invasion. So the solution was to plant trusted settlers from England, Wales & Scotland in Ireland, in large numbers, to subdue the native Irish and be on hand to deal with any invasion. King James I was a Scot and so particularly favoured his fellow countrymen. Much of Ulster, was heavily settled by Scots. During the 1600s, some 200,000 Scots settled in Ireland representing something like 15% of the entire Scottish nation. They didn’t all come as part of the Plantation. Some settled in the 1640s when General Munro's 10,000 strong Scottish army was disbanded at Carrickfergus after the 1641 uprising, and a further batch came in the 1690s due to famine in Scotland.
As far as the Reivers were concerned, King James I decided to move large numbers of them to Ireland around 1610 onwards. He needed settlers in Ireland and he wanted to get rid of the Reivers from the Borders, or at least stop the criminality by breaking their control of that area. So moving them to Ireland was a bit of a masterstroke which killed two birds with one stone. So if your ancestors are Reivers, they probably arrived in the first 20 years of the 1600s, as part of the Plantation of Ireland.
There are no records of individual settlers at that time. We do know the names of the big landowners (Scots & English) but not of their tenants and others who followed them. DNA might get you back to Scotland but there are no comprehensive paper records in Scotland or Ireland which record individuals names or arrivals.
If you visit Border towns like Selkirk, Hawick, Galashiels & Jedburgh, you will find a strong Reiver tradition with folk & food festivals every summer. You can go for a Reiver walk by moonlight and have a 16th century banquet. The usual nonsense. For example, see:
http://www.hawickreivers.com
http://www.borderreivers.co.uk/Border%20Families/BORDER%20SURNAMES%20%202.htm
Armstrong is I think the second most common surname in Co. Fermanagh after Maguire. Their ancestral home in Scotland was Whithaugh Castle in Liddesdale, Roxburghshire and their chief lived nearby at Mangerton:
http://www.gatehouse-gazetteer.info/AP/whithaugh.html
If you want a detailed read about the Reivers, a good book is Godfrey Watson’s “The Border Reivers” published in 1974, ISBN 0 709 4478 4. Plenty of bloodthirsty and cruel tales. The Armstrongs get regular mention.
One of the most famous songs in Ireland, after Danny Boy perhaps, is the Parting Glass. A lament performed here by Liam Clancy & Tommy Makem. However it started life as “Armstrong’s Goodnight” and was reportedly written by Sandy Armstrong, on the eve of his execution in Scotland in November 1600 for murdering the Warden of the Middle Marches (an area of the Borders).
https://www.youtube.com/watch?v=KqcdTinjKvA
