I discovered a few items and have been updating the family tree on the free website familysearch.org.
1. Patrick Carroll's wife was Mary Dane. This is her name listed on Patrick Carroll's Death Certificate, 27Jul 1867 at age 69. Source: "Ireland Civil Registration 1845-1913" My Glascow cousin knew Mary's name was either Dane or Dare, so there is a very good chance this is the correct Patrick Carroll, the father of Michael Carroll.
2. I found the church record for Micheal Carroll and Bridget Murphy's marriage on 14 Feb 1867. Her last name was misspelled as "Murphey" in the database. The extract on Ancestory lists Bridget's father as Cornelious Murphy. My cousin had his name as "Cormack" which might have been his nickname. He was married to Mary Fagan. I found a daughter Hannah's marriage record in 1844 listing Cornelious as a widower. But, I have not been able to find a death record for Mary Fagan.
3. The 1841 Irish Census lists Cornelious Murphy and James Murphy is listed as an "applicant". I do not know if James is his son. Their address is Kellevy, Armagh, Ire.
I have not been able to find any additional census information.
I have also not been able to find Cornelious's birth record, or Mary Fagan's birth record.