This took a great deal of time.
I research the Irish Family History Foundation's database. It contains 12,000,000 surviving rcds for abt 65% of Irish counties. Rcds are increasingly sparse B4 1840, rare B4 1800.
Note 1 - Without a KNOWN county, the chance of an ancestor's rcds NOT being in this database is AT LEAST 35%.
Note 2 - Only 1/2 of Co. Cork's rcds are in this database.
Note 3 - The RC church kept control over most parish rcds from Tipperary (north). These are ONLY available thru the Tipperary Family History Research center at a substantial fee (regardless of outcome).
For Kate Fitzgibbon B 1841 (exact), Bulgaden, Limerick, with a father forenamed James, I found 1 rcd.
Possible siblings to Kate are Bridget 1855, Ellen 1850, John 1846, Maria 1858, Margaret 1864, and Pat(rick) 1844.
For James Fitzgibbon M 1841 (exact), Bulgaden, Limerick, I found 1 rcd.
For Mar(ia) Sullivan M ditto, I found ditto. These rcds are HIGHLY likely to be for your ancestors. A view of the M rcds SHOULD show their parents' names and other info. If it exists, this info will help you identify their B rcds.
For James Fitzgibbon B 1826, Co. Limerick, with the father's forename unknown, I found 1 rcd in Shanagolden parish. The father's forename was Daniel.
For Mar(ia)Sullivan B ditto, I found ditto. The father's forename was Patrick. A view of James and Maria's detailed B rcds SHOULD show their parent's names and other info.
For William Quain M 1867 +/- 4 yrs in Co. Limerick, I found 2 rcds: 1865 Limerick and 1869 Clare. There are NO M rcds for this name in 1867 anywhere in the Ireland in this database.
For William Quain B 1839 (exact) in Co. Limerick with the father forenamed William, I found 1 rcd in Croom parish.
For William Quain B 1811 (exact), Fedamore, Co. Limerick, with the father's forename unknown, I found 1 rcd. The father's forename was Pat(rick).
For Ellen Fitzgerald B 1823 (exact) somewhere in Limerick with the father's forename unknown, I found 2 rcds. The parishes were Pallasgrean & Templebredon, and Rathkeale. The father's forenames were William and Patrick respectively.
Google the Foundation's name to lead you to this database. Register, opt to be notified as rcds are added and browse. Pls post your search results to my genforum msg.
Hope this helps. Good luck.