I searched the Irish Family History Foundation's online database at ROOTSIRELAND.IE. It has abt 20 M B/M/D, census, civil, etc records for MOST Irish counties.
For Francis Sleeth B 1830 +/- 10 yrs somewhere in Ireland, including Co. Armagh, I found 0 B records. The surname Sleeth doesn't seem gaelic-Irish, and hence protestant. Protestant records, especially at this early date, are the least likely to have been recorded and/or survived.
For Bridget McMahon, Co. Clare records haven't been fully transcribed into the database yet. Suggest you visit the IFHF website, register (free), and opt to be notified as records are added to this database.