Good morning,
I apologize if this is information you already have.
There are fragments of the 1821 census, but I do not find a Denis Coffey in it. There are other Coffey families, but they don't look like his family and do not include Denis or in Loughrea. I also looked for a Catherine Flanigan in Loughrea and in nearby towns in the vicinity of Loughrea, but had no luck for a baptism.
One suggestion would be to take a look at the baptismal sponsors and see if they are somehow connected to the families of Coffey and Flanigan. Usually they are. Unfortunately, I did not find a marriage. If they were married in Loughrea it should be recorded as information for marriages is available from 1786. Most likely the couple would have been married in the bride's parish, though, and we don't know where that is.
Denis Coffey, b. 05 Jan 1814, baptised 07 Jan 1814, Loughrea, Parents: Patrick Coffey and Anne Donnell; Sponsors: Patrick Hannin and Bridget Hannin. I cannot be sure that this is the correct Denis Coffey, but I would look into the Hannin family.
There is a sibling for Denis, Michael Coffy, b. 8 Sep 1822, baptised 09 Sep 1822, Loughrea. Again the parents are Anas[t}asia McDonnell and Patrick Coffey, Sponsors: Francis Moakley and, unfortunately, not recorded
The children you have already named:
James Coffey, b. 03 Nov. 1841, baptised 05 Nov 1841. Sponsors: Patrick Kelly and Winifred Kenny
Patrick Coffey, b. 05 Oct 1843, baptised 07 Oct 1843, Sponsors: Timothy Corliss and Anne Flanagan
Ferdinand Coffee [Coffey] b. 08 Jun 1845, baptised 10 Jun 1845, Sponsors: John Corles and Mary Connell
Anne Coffey b. 25 Jun 1850, baptised 18 Jul 1850, Sponsors: Edward Commins and Julia Torpy
Sorry I couldn't be more help. If you have additional questions, please let me know.
Marsha Thomas