There were two seperate St. Catherine's churches and parishes in Dublin city - a Church of Ireland St. Catherine's on Thomas Street and the Roman Catholic St. Catherine's on nearby Meath Street. Many of the parishes named after saints were used by both RC and Cof - e.g. Andrew, Mary, Nicholas....
Most Catholic parishes did not keep death/burial records in the 1800s, and many particularly those in the cities and towns, did not have their own attached graveyards. If you draw a line from Dublin city to Sligo, then Catholic Parishes north of this line are a little more likely to have death-burial records.
The only other reason I've come across is that a burial service did not count as a sacrament, whereas baptisms and marriage do - and the registers were usually considered registers of sacraments. A few registers also recorded confirmations which is also a sacrament ..
Historically many burials took place in old shared graveyards, dating from before the Reformation. The old Established Church (CofI) graveyards were also used by other Christian Denomination - e.g. Prebyterian etc. The non-Church of Ireland burials were often not recorded in the register. Independent cemeteries like Glasnevin (aka Prospect) were different as they were not attached to a church or parish, and in Glasnevin's case it was non-denominational - "for the burial of those of all religions and none". The fact that someone is a widower might be recorded in Glasnevin's burial register, but as far as I now the search function only applies to the first name/surname/age and address. On the free Glasnevin search you can sometimes figure out addresses by guessing a letter at a time - like the 'hangman' word guessing game, which can help narrow down possible matches if you know where a person might have lived.
Most Church of Ireland Burial registers do include records of burial, but of course not all these survive, and many of those that do are not currently available online.
