The couple married before the start of statutory birth registration in Ireland (1864) and so you won?t find a birth certificate for any children born before that. For that you need to rely on church records, where they still exist. You don?t say what denomination your family were but I?d guess Church of Ireland, based on Andrew?s denomination in the 1901 census.
Dunkineely is in the parish of Killaghtee. They have the following early records:
Baptisms, 1810-31; marriages, 1814-31; burials, 1819- 22; marriage banns, 1809-14.
Then there?s nothing till 1873. (The missing records were probably destroyed in the 1922 fire in Dublin).
To analyse whether there?s any connection with the family you have found in Griffiths, we need to know what Andrew Coulter?s occupation was on his son?s marriage certificate? Was he a farmer? If he was, then you might expect to find him in Griffiths and the tithes. However if he was a weaver or labourer, or someone in a trade where you wouldn?t have any land, then he won?t normally be listed. Also what were the couple?s townlands on the 1857 certificate? (I can see that the children were born in Dunkineely but is that also both their addresses in 1857?)
Note that in Ireland Kilpatrick and Kirkpatrick are the same name and interchangeable. I do see a couple of Kirkpatrick households in Griffiths for Dunkineely.
There?s a couple of deaths in the area that might relate to your family. You might want to order copies to see if they shed any information:
Andrew Coulter registered Donegal town, died July ? Sept 1890, est year of birth 1805. Also sane name died Jan-Mar 1888, est year of birth 1795. And 2 Mary Coulters, again registered Donegal town, died 1875 born 1809, and died Oct ? Dec 1892 born 1800.