I just read your information on the link to World Connect. There is a lot of conflicting information. I think your search needs to be focused more on Ohio and on how and when the family came into the US. If it is clear Daniel and Ellen emigrated together, they should show up together on a ship manifest. Have you found one? The 1900 census shows that Daniel emigrated in 1887 and Ellen emigrated in 1887, but they could have come over on separate ships.
I tried searching
http://www.castlegarden.org, but without success. What was their port of entry? Could they have come through Canada? When you can establish a firm immigration path into the country, you have a better chance of finding family records elsewhere.
The 1900 census shows the children Maggie, Daniel, Garrett, and Ivea as born in Ohio, only Thomas was born in Wales. However, your relative? Terry stated that Thomas, the oldest child, was born in Ireland, around 1886. Census records are not error-free, and Terry could be right.
I do not know whether children immigrating to the US, in the late 19th century, had to file naturalization petitions as they got older. (I am aware of cases earlier in the 19th century when they had to do so, and they also had to renounce allegiance to former countries.) I would ask about a search of Ohio records for a naturalization record for Thomas or his father Daniel, having lived in Youngstown, Mahoning County.
http://www.ohiohistory.org/resource/archlib/natural.htmlThe naturalization records might state what ship they immigrated on and when, however, that applies more for records after about 1900. That is why I am asking whether the child Thomas had such a record.
From the information it sounds like Daniel and Ellen were married in Wales maybe around 1883. Have you found a marriage record there?
If Terry is correct about Thomas's birthplace, Thomas's birth should be in Ireland civil registration. Here are some possible matches:
https://www.familysearch.org/search/records/index#count=20&query=%2Bgivenname%3AThomas~%20%2Bsurname%3ADennehy~%20%2Bbirth_place%3ACork~%20%2Bbirth_year%3A1883-1886~&collection_id=1408347However, it is best to work backwards.