I research the Irish Family History Foundation's database. It contains 12,800,000 surviving rcds for abt 70% of Irish counties. Rcds are increasingly sparse B4 1840, rare B4 1800. Cos. Donegal and Derry were recently added to the database.
Note 1 - Without a KNOWN county, the chance of an ancestor's rcds NOT being in this database is AT LEAST 30%.
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).
Note 4 - Microfilms of parish church rcds are kept by the National Library of Ireland (NLI).
For Ros(e) Kilroy M ????, somewhere in Ireland, I found 4 rcds: 1838 Westmeath, 1847 Sligo, 1848 Kildare, and 1900 Roscommon.
For Michael Fallon M in the same counties found above, I found NO matching rcds. They may have married in America.
For Rose Kilroy B ????, somewhere in Ireland, I found 6 rcds: 1862 Roscommon, 1870 Westmeath, 1879 Roscommon, 1891 Mayo, 1897 Mayo, and 1898 Mayo.
For Michael Fallon B ????, Co. Galway, with a father forenamed James, I found 4 rcds: 1855 Rahoon RC, 1881 Gort CP, 1881 Kilbeacanty RC, and 1885 Rahoon RC parishes/districts.
For James Fallon M ????, Co. Galway, I found 23 rcds: 1819 Rahoon RC, 1831, 1838, 1843, 1845, 1848, 1848, 1849, 1858, 1859, 1864, 1866, 1866, 1868*, 1868*, 1868*, 1869, 1877, 1883, 1893*, 1893*, 1893*, and 1893*.
For Mary Connaughton M ditto, I found 24 rcds: 1814, 1819*, 1821, 1822, 1831, 1839, 1844, 1853, 1854, 1855, 1857, 1861, 1865, 1868* Woodford CP, 1871, 1871, 1871, 1873, 1873, 1880, 1893*, and 1901.
Only the asterisked items are for POTENTIAL matches. These must be evaluated at the parish level to see if there is a M match in the same parish. This evaluation showed NO matches. It appears their M rcds didn't survive, haven't been digitized, or your Galway locus is incorrect.
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.