Did the Mayflower passengers know much about the failed Popham Colony of 1607? Would they only know minor details about that colony, or would there have been more extensive study into its location, techniques, tools, etc?
Conversely, what did the Jamestown colonists know about Roanoke, other than the fact that its inhabitants disappeared?
Did either Plymouth or Jamestown change their courses, approaches, etc, based on these prior failures?
I have done a lot of reading into the subject of the Mayflower's voyage and passengers but I recall/can't find any sources that would indicate if the mayflower passengers knew about either colony, however both John Smith and George Percy of Jamestown both have some tid bits pertaining to Roanoke which I have found.
The following account comes from George Percy's account of his expedition into the interior of Virginia.
"At port Cotage in our Voyade up the River, we saw a Savage Boy about the age of ten yeeres, which had a head of haire of perfect yellow and a reasonable white skinne, which is a miracle amongst all savages."
This has been used by many historians as a piece of evidence showing the Roanake colonists possible diffusion into the Native American culture. Percy makes no more comment on the matter and it is just a short note that he makes. This doesn't really prove his lack of knowledge on Roanoke or his knoweldge of the lost colony, but it is a point worth bringing up.
John Smith on the other hand references the lost colony a few times in his history of Virginia, telling about how the natives spoke to him about a group of "certain men cloathed at a place called Ocanahonan, cloathed like me". He also later says that he sent a group of men to try and find these men, but was unsuccessful in doing so. Smith never writes that any of the information of Roanoke influenced him in any way so it would seem unlikely that the colony changed their course on this.
To answer your question directly, it would appear that the Jamestown colonists were aware about Roaoke and perhaps a few minor details about the purpose of the colony.
Sources:
The Jamestown Project By Karen Ordahl Kupperman
Roanoke Island, the Beginnings of English America By David Stic
The lost colony of Roanoke : its fate and survival / by Stephen B. Weeks. New York : Knickerbocker Press, 1891
Jamestown Narratives: Eyewitness Accounts of the Virginia Colony, the First Decade, 1607-1617
Also a very interesting research document I found that shows the relation of Jamestown to Roanake via maps from the period:
Hidden Maps, Hidden City: The Jamestown Connection To The Lost Colony by Fred Willard And Phillip McMullan and Research Assistant Kathryn Sugg
Also if you would be interested in more comprehensive and shortec book about the Mayflower and the struggles in early Plymouth , I highly reccomend Mayflower by Nathaniel Philbrick.
What is the current Historical consesous on what happened to these colonies? Will we never know? Is it still a "Mystery" as it portrayed in popular culture? or is there a mundane answer that is discounted because it is not exciting enough for the conspiracy buffs?