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https://irelandxo.com/ireland-xo/message-board/calhoun-search
What’s the source of your information that Robert Calhoun was born in “Derry, Co. Cavan”? I ask because Calhoun and the variant spelling Colhoun seem pretty well unknown in that County, never mind in Derry. You should also be aware that there are 3 townlands named Derry in Cavan. I have checked the tithe applotment records c 1830 and Griffiths Valuation (1856) and there is no-one of that surname to be found in any of them. Nor anywhere else in the county. No Calhoun/Colhoun in the 1821 census fragments for Co Cavan either. Makes me wonder if he really was born in Co Cavan. The name is more common in Co Londonderry/Derry and Co Donegal.
According to MacLysaght’s “The Surnames of Ireland” Calhoun and the more common version Colhoun are, in Ulster, normally descendants of the Scottish clan Colquhoun. So that would point to them being descendants of Scots who moved to Ireland in the 1600s. They are likely therefore to be either Church of Ireland or more probably Presbyterian. (Presbyterianism was brought to Ireland by the 200,000 Scots who arrived in the 1600s).
As Roger has mentioned, many Church of Ireland records were lost in the 1922 fire in Dublin, and of those that have survived, many are not on-line. Likewise only some Presbyterian records are on-line too, and not all go back as far as the late 1700s and early 1800s. So tracing the family may be challenging. If you want to check out the Cavan records, you would probably need to go to PRONI (the public record office) in Belfast and look them up. Given that there’s 3 parishes involved (Shercock, Crosserlough & Killashandra), that’s probably 6 or 7 sets of records. Where records exist, they should be on microfilm in PRONI.
I wouldn’t worry too much about James exact birth year. Most people in Ireland in the 1700s and 1800s didn’t celebrate birthdays and often had no idea of when they were born. Ages were therefore often just guesses, especially on gravestones. The normal advice is to search 5 years either side of the apparent event date.
I can’t really help with identifying the sister without some idea of her name and where she lived. But if you find James’ baptism, she and any siblings could be in the same set of records.