Just a quick note - there's a probable match for a burial of your Joseph O'Neill in Glasnevin Cemetery (Prospect) in 1902, reported age 52 and the address includes the words 'city' and 'quay' so fits with that death registration at Sir Patrick Dunn's Hospital. If it's a family plot there may be details of others in the same grave ... might be worth following up...
The Glasnevin Cemetery records are pay-per-view at dctrust.ie (Dublin Cemeteries Trust) - there's an extended option to show all those in a grave. Sometimes grave were shared so there could be others .. but if the family owned the plot all the burials in it could be family members
Based on your information I've seen the death records for Joseph, which includes his home address - they dont always do this ... so lucky break there, also the census for him and his family in 1901 and the couple marriage. Do you have 1901 census details for Mark & Francis ?
Death records are the most difficult source to match up, or rule out, the only real clues, besides name and age (which is often inaccurate), can be occupation, address and informant ..
A quick search of deaths of a William O'Neill (and variants) from the start of civil records in 1864 up to 1905, with a year of birth around 1860 in Dublin city and county shows about 20 matches - register images for those from 1870 are on the IrishGenealogy website. If 'your' William died before the start of civil records there would be no death record but there could be burial record if his grave was in Glasnevin (which opened in 1832).
I did a quick search of the burials and found a William O'Neill burial in 1885 with 'city' mentioned in his address, a cross check for deaths located a matching civil death record - William O'Neill died 17th June 1884 at Sir Patrick Dunn's Hospital (see entry 426 in the register image) noted as from 22 City Quay, reported age 22 occupation a labourer, single - the informant is a W. Hiddle, for the Hospital.. looks like a pretty good match - same street, age is very close, even the same place of death as Joseph...