I came across this dialogue while doing some research on the Brennans of Lemgare, Clontibret parish, Monaghan County, in anticipation of my first visit to Ireland from the U.S next week. QueenG: it seems that you're looking for information regarding the same family history as I am. I'm a descendent of Denis Brennan, brother of Mary Alice, born in 1864. My brother did an extensive genealogical research project in 1993 to uncover what he could about our family heritage before visiting Ireland himself. He, too, could not find birth certificates or church records at that time for most of Denis' siblings, who were the offspring of George and Mary. I know that this is the same family as you are looking into, because my brother was able to determine that Mary was also a Brennan prior to marrying George, but we don't know how that relationship played out and if she was from the lineage of Brennans from the Lemgare Valley or not.
We do know that the siblings of Mary Alice, who you're interested in, were actually Michael (Mick), Bernard (Barney), my great-grandfather Denis, Patrick, Bridgette and Kate. We don't have any record of a sibling named Sheila. Bridgette married Michael MacCormack in Castleblaney, and most of their descendants apparently remained in Ireland, mostly in or around Monaghan County. Patrick married Mary McNancy and had six children, one of whom was Sheila. That might possibly be the Sheila you referred to. Denis emigrated to the U.S. in 1885 via New York and settled in Rochester, New Hampshire, where he married and raised seven children. My brother did find out that Mary Alice married Patrick Traynor from Tullycorbett. We are aware of one son of the Traynors, Tom Traynor, who informed my brother that his mother and father are both buried in the family plot in the Tullycorbett parish cemetery. Unfortunately, my brother was unsuccessful in locating their gravestones during his visit. My brother was able to locate another Mary Alice Brennan, of Glaslough, Monaghan County. "Mary Alice II", however, is the granddaughter of Patrick Brennan, brother of Mary Alice Traynor, nee Brennan.
Adding to this mystery is the fact that the other sister of Denis, Kate Brennan, married a man whose last name was Nugent from the village of Darkley. We also know that Kate never had children. So, after this long-winded discourse, it seems that the Patrick Mulligan you're interested in is probably not a descendant of this Brennan lineage, unless there was some serious misreporting of births and marriages at that time, which is possible. You're likely aware that until around 1861 in Ireland, there were few official records kept of Irish marriages and births under British rule. My brother found this out while doing his research, based mostly on what he could find in the Irish Civil Records files kept in the Mormon Family History Center in New York.
I hope that this has been helpful to you and would like to hear back from you if you are able to successfully determine the relationship of Archbishop Mulligan to Mary Alice Brennan. All the best.