I have just returned from the National Archives in Montreal today, and am on the very same trail as you with the same questions you are asking. I am a direct descendant of the John Edgar and Mary Arken union, as they would be my 3x great grandparents. Not listed among the many progeny (and many wives) of John Edgar is his first born, my 2x great grandfather, Thomas H. Edgar. He was born in Quebec City on 28 October 1829. He died on 13 July 1913 in Baltimore. His death certificate clearly states his father John's birth in England and his mother Mary Arken's birth in Ireland. I could not locate his baptism record unfortunately; perhaps the recod has been lost or that he was baptised in another parish other than the one that John and Mary were married on 3 Nov 1828.
I will go on the trail for John Edgar's origins in Chatham, as I have a Hampshire geneaologist on board who has helped me with one branch of my family (Herbert), who married into the Thomas Edgar family.
We are obviously half cousins. To quickly summerize, my Thomas Edgar moved to New York just prior to 1850, as he appears in the 1850 census as an umbrella maker. He married my 2x great grandmother, Mary Ann Herbert, in May 1853. Mary Ann Herbert arrived in the US in 1844 with her widowed mother from Portsmouth, Hampshire. While this Edgar branch has not survived sucessfully in terms of the surname being passed on (save one small branch), be glad to know that there are many descendants by other surnames of the Edgar-Herbert union - the vast majority of them Moultons, the maiden name of my Grandmother Morse.
I would be delighted to make contact with you, so please feel free to contact me directly at email@example.com. I would be more than happy to share further information, and equally so, to know from you who the many Edgar cousins are and where they currently reside. I would especially enjoy seeing an image of John Edgar, should one exists, as he lived long enough to be a part of the photographic age. I grew up along the Hudson River north of Manhattan but have been living in the Boston area since 1983. I look foward to the possibility of hearing from you. Sincerely, Christopher G. Morse