Hi -
Are you sure that Edward is the oldest child? What did Catherine say in 1900 and 1910 for "how many children, how many living?" They were probably married within a year or two before the oldest child's birth. Another reason I ask is because, if those are all their children, then they didn't name any of them after their parents.
Can you find birth records for the children? Do they give the address where they were born?
https://a860-historicalvitalrecords.nyc.gov/ They were probably attending the nearest church to where the child was born. Most children were born at home.
Catherine O'Shea couldn't marry until she arrived, so you can cut that immigration window down by two years. The passenger manifests for that era almost never tell you anything about place of origin, but still worth a look. Every so often I have seen a manifest that includes a more specific location than just the country. (But maybe at a different port of entry, I don't remember.)
If William arrived at age 13, then he arrived with parents, or at least with family. Can you find them?
Catherine much more likely to be on her own - 18 to 20 is when you'd leave to find work.
Are any known relatives in the same area of NYC? If so, work on their families as thoroughly as if they were your own.
Any boarders or lodgers on the censuses named Sullivan or Shea? City directories are tough to work with, but if you find your William in one, are there any other Sullivans or Sheas at that same exact address? If it were a rarer name, I'd say same street, but not with Sullivan or Shea.
does the DNA point to any specific areas of Ireland?
There were an awful lot of Sullivans in Cork. but not many Simpsons.
You can put two surnames into this site, one at a time, and see where they overlap. But get it right the first time because he only allows one search per day, I think. A different browser might work but it looks like he might have tightened that up also since I tried it last. I wish I had taken a screen shot of the results for you because I can't repeat the search.
https://www.johngrenham.com/surnames/edited to add:
The second set of results at that site shows you the overlap of the two surnames. If I remember right, there were a lot of places where they both show up, so not a lot of help, but worth a look
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Simpson is the unusual name in the bunch.
Who was the informant on William's death record? Not Catherine who'd be the most likely to get both parents' names right. Could the informant have got William's father's name wrong? It has happened. And I've seen the "replacement" name be the deceased own name or his oldest brother. Like the informant took the best guess he could.
But "Simpson" is a name you wouldn't forget. I'd bet more on that being correct.
The reason I ask is because I looked at familysearch for any Sullivan born in Ireland 1850 to 1870 with father named __ Sullivan and mother: Sarah Simpson
There are three records for a couple Jeremiah Sullivan and Sarah Simpson - births in Bandon, Co Cork 1864 to 1868.
and a marriage of a Maria Sullivan b 1855, married NY County in 1881
https://www.familysearch.org/ark:/61903/1:1:24S3-9L7?lang=enhttps://www.familysearch.org/en/tree/person/details/GYJ5-BB1I'd would want to know who Maria was.
try the same method - widening your search - at
https://search.findmypast.ie/search-world-records/ireland-roman-catholic-parish-baptisms and elsewhere.
I don't value myheritage highly. I haven't seen any records there that aren't at ancestry. They say on the Irish forums that rootsireland.ie is the best for Irish research. But you're not there yet solidly enough to justify a subscription right now, I don't think.
Findmypast will let you at least search for free, as Shelly mentioned. I don't know about rootsireland.
https://www.irishgenealogy.ie/ has the civil records for Ireland, but they don't start until 1865 except for Protestant marriages - forgot when they started. But I'd bet heavily that Sullivan - O'Shea were Catholic.
Still you might find later siblings on that site, if you get to that stage.
edited to add
And there is also the possibility that William and Catherine's families lived within walking distance of each other in Ireland. People went where they knew people.
