Neil Diamond : Short Biography

Hello, I am a quarter Irish in my heritage, and it seems to be the hardest part of my heritage to find. The Surname I am looking for is DIAMOND:

The first is a Neil Diamond, who was born roughly the year 1776. He and his wife (known only as Rose) had what I believe to have been the following children while in Derry: George (my ancestor) born c. 1800, Henry born c.1812, Bridget born c.1814, Martha born c.1816, Mary born c.1818, Neil born c.1820, Charles born c.1822. These individuals traveled together in June 1840 out of the port of Liverpool, to the Port of New York. Once in New York they settled on Manhattan's lower east side where sons George, Henry, and Neil all married and brought forth future generations. As to the son Charles, he died young in America, and I have not been able to trace any marriage or death records for any of the daughters. Here in  America, (New York City specifically), I also found a Daniel Diamond (of Irish birth) who was a noted witness to the wedding of my ancestor George Diamond to his wife Margaret Kane who was noted as being from Co. Cavan. Due to the early record keeping traditions of when Neil Sr. and Rose died in NYC, there is no information as to their parentage. This family was Catholic in faith.

The second is a Martha Diamond who was born in Derry, Ireland  in the year 1825, and traveled to Glasgow (alone or with family has never been found) where she married to a John Cairns of Co. Antrim. There in Glasgow they had several children before eventually making their way to America in 1856/1857 (first John, and them Martha with their children). Once here, several of their children died right away, with the exception of their daughters Anne born in 1850 and Sarah born in 1855. In 1869, their daughter Anne married to a George Washington Dimond (son of the previously noted George Diamond and Margaret Kane). That is where my continued ancestry lies. This family too was Catholic in faith.

So... yes I am curious to know if anyone out there would be familiar with any of these names, and if so, is there any further available history for the Diamond family. It would also be interesting to see if Anne Cairns and George Washington Dimond were distant cousins through Anne's mother, Martha Diamond. 

One last question: a tough one... were the Diamond clan of early Celtic stock; no matter where I look, they seem not to have descended from any major Irish Clans of Chief Septs. How far back into Ireland's history do they really go preceding the time of The Plantation?

Any bit of insight is most highly valued, so thank you in advance.

Quarter Irish.