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 Mary Molloy B 1872 (exact) somewhere in Ireland with the father's forename unknown, I found 24 rcds. Four are in Co. Dublin. Note - County Dublin rcds are in the database, Dublin City rcds aren't. Since the four rcds appear at the start and end of the list, none are likely to have a May DOB.
It would have been unusual for a child of 14 to emigrate in 1886 alone. Generally the unaccompanied age was at least 18 for women. Many were accompanied by a close relative, or family friend. They often emigrated to join extended family members already in America. This would be shown on the immigration rcds when she disembarked, likely in New York city. Google "castlegarden" to find emigration rcds prior to the opening of Ellis Island in 1892.
Also try to see the marriage rcds at St. Joseph's. These often record the parents' full names and sometimes place of origin. B/M/D civil registration was beginning in the U.S. by 1890. PA being one of the more progressive states at the time, may have required the pair to pay a fee and fill out a marriage license. These also rcd'd essential info.
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.