Thanks for the colorful family history. I had an aunt who was a lovely flapper.
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 Rose Kilroy B ???? somewhere in Ireland with the father's forename unknown, there are just 7 rcds: 1846 Tipperary, 1862 Roscommon, 1870 Westmeath, 1879 Roscommon, 1891 Mayo, 1897 Mayo, and 1898 Mayo. NONE are in Galway.
For Michael Fallon B ???? ditto with a father forenamed James, I found 44 rcds with 4 in Co. Galway.
Suggest you find reasonably close DOBs, parent(s) name(s), and/or placename(s) to pick which, if any, rcds are for your ancestors. Censuses and naturalization papers are good starts. Also, Social Security Death Index (Google SSDI), WW1 draft registrations, civil B/M/D rcds, etc are invaluable.
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.