Hi, just wondering if the father's name should be John rather than James ?
The births for Terresa and Emalia both give their father's name as John, occupation RIC Constabulary. There's a good match for this John in RIC service records as Jno. Cullinan, county of birth Clare, signed up 1838, age 23 (b1815). His service number is 3198, and it seems he only served in Co. Mayo, the record also mentions that he is Catholic and married 21st October 1847, wife also born Co. Clare. He was pensioned in June 1873.
RootsIreland has a transcripts of two baptisms for a Catherine Cullinane/Culanan in 1848 and 1856, both Catholic and in Co. Mayo with parents John Cullinane and Mary Tighe - I have not been able to verify either of these in the NLI register images as yet.. if these are correct then it would appear that the first Catherine died young and they reused the name. Do you have an approximate year of birth for your Catherine ?
John & Emily married in Westport on the 8th October 1893, and son John Joseph was born James' Street, Westport on the 2nd March 1896. He seems to be listed as Joseph on the 1901 census, and John Joseph in 1911. Marriage records on the free IrishGenealogy website cover up to 1941, so if he married somewher in the Republic of Ireland there should be a record in these.