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indexed in 1882 but the date of the certificate says 1881. either the minister or the indexers got it wrong.
Robert Rountree and Maria Sullivan
married 8 February 1881 by Thomas Henry Lee, pastor, St Christopher's Church. his residence: 204 West 39th
witnesses James Brown and William Cairns

Robert Rountree of 227 West 40th, age next birthday: 27, a coachman, born in Ireland,
Maria Sullivan of 27 Beekman Place, age next birthday: 26, born in Ireland
first marriage for both
parents as named in the last post.

https://a860-historicalvitalrecords.nyc.gov/view/11977894

son William's birth
at 246 West 36th
19 January 1886
Myrae/Maria's age 30, born in Ireland

https://a860-historicalvitalrecords.nyc.gov/view/282502

sons John's birth
at 304 West 22nd
Maria's age not given, born in Ireland
the father Robert was a coachman on both birth records
https://a860-historicalvitalrecords.nyc.gov/view/138528

haven't found son William's death yet.

Regarding the religion question, I looked at the marriages of the children.

That might have done it.
Protestant marriages were recorded earlier.

Are these his parents?
Marriage record for Jeremiah Sullivan and Sarah Simpson
Group registration ID: 3216521
SR District/Reg Area: Cork

Party 1 Name:    Jeremiah Sullivan
Party 2 Name:    Sarah Simpson
Date of Event:    07/10/1851
Group Registration ID:    3216521
SR District/Reg Area:    Cork
Image path:    marriage_returns\marriages_1851\09407\5407823.tif

Parish Church, Parish of the Holy Trinity, City of Cork
October 7, 1851
Jeremiah Sullivan, of full age, bachelor, teacher, residence: Queloyne?, father: Michael Sullivan, a gardener
Sarah Simpson, of full age, spinster,  teacher, residence: Morrison's Quay, father: John Simpson, a pensioner
she signed with her mark
married after banns by Robt Maguire [can't read his degree or id. it looks like CLR - clergy? clerk?]
witnesses Ralph Hodges and Denis O'Connor

https://www.irishgenealogy.ie/view/?record_id=8f6abc883a-2038174

still working on Jeremiah's residence

There were two other births that I wondered if they might belong to this couple.  William Sullivan and Catherine/Katherine O'Shea as parents, William's occupation as shipping clerk fits with the other births, and one is in the Greenwich Village area and the other is also in lower Manhattan. 

Anna Sullivan, b. 25 Nov. 1895 https://a860-historicalvitalrecords.nyc.gov/view/958823

Elizabeth Sullivan, b. 23 May 1891 https://a860-historicalvitalrecords.nyc.gov/view/556706

So Edward was the oldest child, since those records list number of children and number alive. Unless there was a miscarriage or stillborn child that she wasn't counting.

I looked for the marriage at https://www.italiangen.org/ and didn't find it. I looked for all 5 boroughs from 1882 to 1886.

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1891 City Directory
3 men named Wm Sullivan, clerk
at  890 Boulevard
at  1384 3rd Ave
at  115 Washington Place

1892
Wm Sullivan, clerk 18 Beach
others at   
  890 Boulevard
  1384 3rd Ave
  122 E 113th
  311 W 13th
 
assuming that is him at 18 Beach in 1892, then that eliminates 2 of the 3 addresses in 1891, because those men are still at the same addresses, leaving 115 Washington Place as the only possible address for your William in 1891. - assuming he was listed in the first place. You can work backward and forward that way.

I didn't find any other Sullivan or Shea or O'Shea at 18 Beach in 1892. I didn't check in 1891 since I did that year first but wasn't sure which was your William.

Close enough to to take a better look at, if you can find these people:
1892 City Directory
?? Jeremiah O'Shea at 26 Beach
?? Margaret O'Shea, widow, 22 Beach

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John Grenham's site must have a 24 hour time out, since I couldn't do the search today - a new day but less than 24 hours since the last search.
If you get two tries at that site, try crossing Simpson with Shea, on the chance that the two families were from the same civil parish.  It really won't help you know where your people are from, but it's interesting and it might narrow down the haystack in which you are looking for the needle.

But the only real way to do it is to find something on the US records that ID's the birth place.

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stevemorse.org has a way to see the manifest of every ship that arrived. You can narrow it down to July 1873 and see if you find William. I think the start page is one of these, but I'm not positive. I haven't used it in a while.

https://stevemorse.org/ellis/mmfs.html
https://stevemorse.org/ellis/boatfs.html

But there is always the chance that William did not remember the right month or even year. Still I trust the naturalization papers a lot more that whatever they say on the censuses.

There is a William Sullivan who arrived  May 3, 1873 on the SS Denmark with parents Timothy? and Mary, but that is the only one that comes up for that year.

https://www.familysearch.org/ark:/61903/1:1:QVR3-VHYH?lang=en

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=en
https://www.familysearch.org/en/tree/person/details/GYJ5-BB1

I'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.