The cardinal rule of genealogical research is that nothing may be accepted, or reported, as fact, until the person receiving the information has PERSONALLY verified it from the source documents required to be provided by the person giving the information.
That Thomas Culwell was born in Ireland, immigrated from Ireland to North Carolina with a brother, Alexander, and married in North Carolina, is all unproven, and unsupported, legend, exclusive to only a few members of only one of the many branches of the Culwell family. It first surfaced in 1936, as a classroom assignment on family history, done by a Culwell descendent. To date, not one of these claims has been substantiated by any actual researcher.
A member of this branch of the family, a skilled and conscientious genealogist, posted charts on the internet, which state Thomas was born in Ireland. She did though, in her accompanying notes, make clear that this is unverified, and other Culwell researchers believe he was born in North Carolina. She isn't responsible for other people not verifying the claim, and/or ignoring her warning that it's unverified.
If you're familar with census records, and you should be, you know the 1880 census required the people being ennumerated to give the birth places of their parents. If you're familiar with Culwell family history, and you should be, you know four of Thomas Culwell's sons were alive in 1880. If you know what's avaiable on the internet, you know the transcripted census records are easily found. If you'd checked to verify Thomas's birthplace, and you should have, you'd know William didn't answer, but Thomas Jr., Andrew, and James, all stated their father was born in North Carolina. This is documented, first-hand testimony, coming from three different sources. It has to be accepted until, or unless, it's been proven wrong. It hasn't been.
If you're a beginning genealogist, trying to gather information on the internet, stand warned that it's a garbage dump of misinformation. Follow the rules, and accept nothing as fact until you've found the source document, and personally verified it. If you don't, you'll compile a family history that's likely to be far more fiction than fact.