I research the Irish Family History Foundation's database. It contains 12,700,000 surviving rcds for abt 70% 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 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).
For Ellen Connor B 1869 +/- 1 yr, Oughterard, Co. Galway, with a father forenamed John, I found 1 rcd in 1869, Kilcummin Oughterard RC parish. This matches your criteria exactly. It is highly likely to be for your ancestor. A view of the detailed rcd SHOULD show the correct parent's names as well as other details.
For John Connor B pre 1869, ditto, I found 2 rcds: 1863 and 1865. Again, both meet your criteria. It will take a view of the detailed rcds to see which, if either, has a mother named Margaret Logan.
For John Connor M 1864 +/- 4 yrs, Co. Galway, I found 12 rcds. However, only 3 in 1860, 1860 and 1861 were in Kilcummin Oughterard.
For Marg(aret) Logan M ditto, I found 3 rcds: 1-1860 and2- 1864 - 1 church, 1 civil. Only the 1860 rcd was in Kilcummin Oughterard parish. This and the corresponding rcd for John are highly likely to be for these ancestors. A view of the detailed rcds SHOULD show if they married each other. If so, this validates the rcds. The view MAY show their parents' names. This is valuable in validating the B rcds, if any, for the couple.
For John Connor B 1835 +/- 4 yrs, Kilcummin Oughterard, Co. Galway, with the father's forename unknown, I found 1 rcd in 1838. Father's forename was Michael.
For Marg(aret) Logan B ???? Co. Galway with the father's forename unknown, I found 2 rcds: 1829 Kilcummin Oughterard and 1836 St. Nicholas Combined N&E. The father's respective forenames were John and Thomas. The 1836 rcd appears the more likely on the basis of Irish marriage practices.
Will leave it to you to process the above info validate which rcds, if any, are the right ones. With some luck, you may be able to duplicate this process and find the rcds for one more generation.
For Ellen's life in America, she likely died after 1900. B/M/D registration was almost universal after 1900. You are likely to find her death cert. at the California Bureau of Vital Statistics, or similarly named agency.
For Ellen's immigration to America, Google "castlegarden" to find the immigration and passenger lists.
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.