You're correct that we don't know exactly when Andrew arrived in NS. We know he registered himself, wife Sarah, and 4 children in 1765 in the Granville Township Book, so he *probably* arrived that year. He received a land grant in 1765, too.
We don't know who his father was, so I'm not sure why you're puzzled about him not being in his father's will. Did you type the wrong thing in your message?
Andrew and Sarah conceived their daughter out-of-wedlock and were fined in April 1758, but they had been married since January 1758. Catharine was baptized on 21 June 1758 in King's Chapel, Boston, MA - Catharine White of Andrew and Sarah White, infant; sponsors: Timothy Hurley, Susanna Rally, & Judith Lewis. Source: The Records of the Churches of Boston. (Boston, MA: New England Historic Genealogical Society, 2002), pg. 56. I don't know if Catharine was actually born by the April court date or if the court officials just counted the months on their fingers and knew the math didn't work out. :-)