The Church of Ireland parish records for births only start from 1877-1900 and that is why you will be having difficulty getting the records for Tawnagh(Riverstown), there are however, marriage records from 1845-1901 for the Church of Ireland, Protestant. I may be making an assumption here as most early Irish to Upper Canada were Protestant followed later by Catholics.
RC records are kept at the National Library in Dublin but were copied by the Latter Day Saints and
then put on microfilm. They are available on their website www.familysearch.org
Neither place has 100% of records.
After 1864 - Civil Registration began and you will see
deaths of people born as early at the late 1700's as well as birth and marriages after 1864.
General Registry Office, Dublin www.groireland.ie
Tithe Applotments (census substitute) 1823-1837 www.genealogy.nationalarchives.ie
57 Culkin's, and 32 Byrne, 0 Beirn, 4 Byron
In the Griffith Valuation 1846-1864 and 1857 in Sligo.
The only alternate spellings for Culkin is Cuilkien in these records and this is the only 1 listed
Mary Cuilkeen, Parish of Toomour, Townland Brougher, Poor Law Union Boyle
tenant of Mary Breheny
There are 12 Culkin's in Sligo in the following Parish's in 1857: 4 Castleconor, 2 Killoran,
7 Kilmoremoy, 1 Templeboy
I use a membership program for Grifith Valuation and would normally give a good source of Griffith Valuation as www.askaboutireland.ie
(but didn't find any of the spellings of Culkin?) so the only source other than www.familysearch.org
or ancestry.com is: www.failteromhat.com
but it does not provide the details of the property just the Parish and Townland.
You have to be quite imaginative in spellings for Irish names, as a lot of the population could not write
and someone else wrote for them and usually wrote what they knew or by sound.
The only Byrne/Byron in the Parish of Tawnagh:
Patrick Bryon, Parish of Tawnagh, Townland Carrowkeel, Poor Law Union Sligo (within 12 mi)
tenant of Sir Gilbert King
6a house and land 7 acres 1 rood and 35 perches
You are correct that it is possible records could be in Roscommon as some of the Poor Law Union would go across a border, sometimes 3 county borders.
both these have a variety of information depending on the county, some parish records and
There are Culkin's listed in the Sligo telephone book, not too large a number, you could try writing letters to the addresses given which are sufficient for mail delivery and no postal code is required.