I'm a descendant of Mary Peacock and James Arnold through William Arnold, who I believe is their grandson by Jonathan Arnold and Catherine Hicks. I've seen a couple of different names suggested as Mary Peacock's father. Joseph Peacock is the one I've heard most commonly, but I don't know if there's any documented proof of that.
I'm curious as to your statement that Mary was born in Kentucky. On the 1850 Federal Census in Hickman County, TN her state of birth is listed as Tennessee. I've seen others claim that she was born in Georgia, but that might have been based on the assumption that she and James were born in the same state. James is listed as being born in Georgia on that same 1850 census.
My great great grandmother was Mary "Polly" Arnold, who married Joseph Nathaniel Horner. She was the great granddaughter of James and Mary. I found some interesting details on the family, like stories of James killing the last bear in Beaverdam Springs in 1823, and another story I found about how he owned a "tar pole" wagon and temporarily moved his family into a cave after the land he and the family settled on was claimed by another man. Another account described him as an honest man who always paid his debts, albeit sometimes with animal skins instead of coins.
I also came across a post on a website from a descendant of another of James and Mary's children, daughter Mahala, which claimed that Mahala was 1/4 Indian. I don't know which of the parents where half Indian. Peacock is either Irish or English and I've read that this line of Arnolds is Irish. Either James or Mary could have been native American on their mother's side. My mother always claimed that there was Indian ancestry on her father's side and this could very well be where it comes from.
I've had the ftDNA Y-chromosome test that only looks at my direct paternal line. My sister just got the results back on her mitochondrial dna this morning, but I haven't had the chance to see the results, but the Arnold ancestry wouldn't show in that test either. I'm contemplating the Family Finder test, but my understanding of that test is that it shows connections to about 5 or 6 generations back. For me that would be as far back as Mary Polly (Gen 5) or William Arnold (Gen 6). If that's the case then it may or may not be able to prove a connection between our two lines. Many people seem to think that William was a son of James Arnold, but I tend to think he is the grandson, the son of Jonathan Arnold. So that would mean Jonathan, not William is the brother of your G-G-grandmother. Incidentally, I think there are a lot of people on Ancestry.com who've confused the lines of George Washington Arnold and William Gray Arnold with the Arnolds of Hickman County, so don't be confused if any of their descendant's DNA don't match yours. The best I can tell is that George Washington Arnold was from Dickson County, TN, so he's not a likely candidate for James's son George, and William Gray Arnold was alive in Missouri up until 1910 while my William Arnold, William H. Arnold died in 1875. Many people have my William's wife Elizabeth listed as William Gray Arnold's first wife, but she was still alive in Perry County, TN and listed as a widow while William Gray Arnold was still alive and well in Missouri with another woman.
As for the stories of Indian ancestry, it's possible that they are more family legend than truth. Both my parents told of native american ancestry in multiple lines of their family, but to date I've heard only stories and have seen no hard evidence. I've come across one branch of my family that was clearly of African descent, but they were known as Melungeons and many probably perpetuated stories of Indian ancestry. James Arnold seems to have been by all accounts quite a frontiersmen and this may account for some of the belief that there was native american blood in his veins.
I have no immigration information on the Arnolds coming from Ireland. I've read that from other descendants of James, but that's strictly anecdotal and I have seen no proof. It's quite possible that the family was Scots-irish or even Welsh.
I saw that Rhoda was a daughter of David Arnold after I looked and couldn't find her as a daughter of James and Mary. There is an listing of Perry County, Tennessee cemetaries that covers Arnold Cemetary and gives a good bit of information on David. I believe he was the first Arnold buried there and supposedly one of his grandsons by Rhoda is buried there too. The page give a lot of information on David's Civil War career along with his older brother Gideon.
I've taken an autosomal DNA test since we last spoke, through ftDNA (kit # 242755) and submitted it to GEDMatch (kit #F242755). I'd be curious if you ran a comparison with your autosomal results. I'm fairly confident that my William Arnold isn't a descendant of Catherine Hicks, but I think he may be a cousin or nephew of the Jonathan Arnold in your line. I recently matched up with aother autosomal tester and the Arnolds are the most likely link I see between the two of us. He can only trace his line back to Kentucky, but we were estimated as being about 7 generations back to our MRCA, which puts me right about the time of my William Arnold.
I'm interested to learn what connections your Family Finder test made on the Arnold side. I've been researching the family of James Arnold and Mary Peacock, and their son Jonathan and his wife Catherine HIcks for a few months now with the belief that Jonathan and Catherine are the parents of my 3rd great grandfather William Arnold. Their son William H. Arnold is the right age and was in Hickman County at the time my 3rd great grandmother Elizabeth Groves married a William Arnold. The only problem is, I've just discovered another William Arnold in Hickman County about that same time and I'm now uncertain as to which family my William Arnold belongs. If your Family Finder test matches any of the descendants of my William, I'm in good shape. If not, I may need to be tested myself to be sure. Did your test turn up any obvious Arnold descendants? I'd be happy to share my Ancestry.com tree with you to show you my William's descendants in case any of them ring a bell.