Your Katherina Lynam, I think, was the sister of my 4xGGF, Andreas [Andrew] Lynam, who married Sarah Hall 25 June 1751, in New Castle DE.
Both Katherina and Andrew were children of Anders Lynam and Brita [Joransdotter, I think].I have, but have not yet connected,John and Margaret Lynam, who came to Philadelphia in one of the Penn groups in about 1676.They were well-known Quakers, and Margaret was a preacher and writer.A book of religion? philosophy? by Margaret, who also was a preacher, apparentlystill exists and is in the Haverford College Library. Still searching for more on this, but I think the [probably] English John and Margaret probably had offspring, and that one of them probably married into the New Castle bunch which had, in addition to Swedes, Belgians, Dutch, Norwegians, Finns, French and about everything you could think of.
My Anders Lynam and wife Sarah Hall moved with one of the groups of Quakers who went to North Carolina. One of his sons, Andrew Lynam, at age four (maybe as an orphan), went to Virginia, later served in the Revolutionary War as a chaplain, and at some point became a Methodist minister, married Elizabeth [Betsey] Greene, went to Bath County, where he and Betsey had six children.The eldest was Richard, who married Sarah [Sally] Hunt, and then there were some Anderson's who got mixed in and Richard Lee Hezekiah Hunt Lynam (my 2xGG ended up getting married to Mary Jane Anderson, who was also the niece of his mother -- oh well, I could send you all of this in chart form if you like.But anyway, all of these people produced descendants who(some of them, anyway), ended up (via Mercer Co. MO) in Elk and Moline Counties, KS. And one of their descendants, Albert Andrews, is my Dad, who lives in Wichita KS.
All of this is by way of saying, did the offspring of Katherina and Peter Anderson include a Richard Absalom Anderson, b 1805, Morgan Co. KY, or any Hunt's, Greene's, Hall's, Foster's, Doak's?
If any of this sounds likely, please let me know.