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Bridget Ward in 1910 Census, Bridget Rooney claimed to be travelled to her cousin Patrick Ward in 1907, her mother was Bridget Martin, but there is a marriage shortly after she arrived. So was he her cousin or was it a little fib.
https://www.familysearch.org/ark:/61903/3:1:33SQ-GRKL-5WM?view=index&personArk=%2Fark%3A%2F61903%2F1%3A1%3AMG31-ZWN&action=view

https://www.familysearch.org/ark:/61903/3:2:77TH-4FXQ-6?view=index&personArk=%2Fark%3A%2F61903%2F1%3A1%3AKFDP-6N5&action=view

But Bridget Rooney clearly went to America.
So that's Owen, Charles and Bridget.
James is still unknown and we don't know if there were children older than James which had already left home by 1901.
James isn't in the 1911 Census with his mother
https://census.nationalarchives.ie/pages/1911/Louth/Dundalk_Urban_No__3/River_Lane__S__Side/905246/

I know you are hoping to find relations alive today but we can't look for living people on RootsChat the best we can do if possible is find who Owen's brothers and sisters married.
Do you know if he travelled alone to the U.S. or any of his siblings went with him.

Charles died 1912, so no descendants from him.
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_returns/deaths_1912/05359/4493890.pdf
Mother Bridget died 1927
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_returns/deaths_1927/04977/4356187.pdf
Informant was her niece, could that mean that James and Bridget had left the country, they are not at home in 1911
https://census.nationalarchives.ie/pages/1911/Louth/Dundalk_Urban_No__3/River_Lane__S__Side/905246/

Harken is in Pullaghera in the Applotment books, and in Griffith's Horken is in Cloonfinnaun with land next to Patrick Kennedy, the names appear a few times in Griffith's, on Land, houses and bog.

Mary's baptism, with the name as Lavin
https://registers.nli.ie/registers/vtls000632338#page/135/mode/1up
and a new spelling for that place name.