1) Biblical genealogies and ancient king-lists from Egypt and Sumer are alike in their tendendy to list the same person multiple times under different names. A single royal person could be considered the son of more than one father and could be the king of more than one city. In general, the father of a king was his predecessor at a given city/region, regardless if this was strictly true or not. Some Biblical genealogies are in fact just a succession list of kings. Sumerian king-lists were explicitly tied to cities. There was some tendency to do this in Egypt, as well, although the regional affiliation is not always made explicit.
Besides having a political father, royal princes might also have a legal/adoptive father as well as a natural father (as in the Biblical story of Isaac, who was the legal son of Abraham but true son of Abimelech). This part of royal culture allowed for additional creativity in constructing genealogies.
Names from king-lists as well as Biblical genealogies/stories reflected different regional and ethnic identities as well as personal traits. Names were not only considered tags but a part of one's own being. Names were then very important in the ancient world. Biblical writers and composers of king-lists tried to preserve as many unique names and epithets as possible in order to more fully capture a ruler's character and divine fate associated with his assigned role or roles. (We have seen where multiple names of a great king were deliberately interwoven into a single Biblical account.)
I don't know that there would be any formula that could be extracted from a king-list in order to directly interpret another king-list or genealogy. The key that unlocks the history is that within each dynasty the various roles of the pantheon were passed out to members of the royal family. This is the hidden "order" in each list.
2) The rulers of the Middle Kingdom did look to the Old Kingdom and not just the dynasty of the gods for inspiration. For example, the first king named Senusret (Kenan) was intent on being the Khufu of his dynasty. However, in retrospect the author of Genesis denied him that status and patterned him after the murderer "Cain", that is, after the god Anu. Genesis instead makes Senusret II (Jered) an incarnation of "Irad", that is, after the god Re and the later Khufu.
The royal family could not always agree on how the various required roles were to be handed out. Sometimes typecasting was changed in the middle of a dynasty. For example, after the death of Hatshepsut it was decided that her assumed typecasting had been in error and her memory was literally buried. The mother of Thutmose III was thereafter looked upon as the "true Isis" rather than Hatshepsut, and Thutmose III as the "true Horus".
At first, I thought that the Book of Genesis almost completely neglected the Old Kingdom pharaohs. (I called this the "Biblical Dark Age".) But now I see that this was not the case and that there is perhaps only one generation missing in the Old Kingdom genealogy. I am also finding that later dynasties of Egypt had detailed records of the Old Kingdom and used them for inspiration as much as any other dynasty or the gods themselves.
The designations of Old, Middle, and New Kingdom are rather artificial and mainly reflect our modern conception of how Egyptian history played out. The author of Genesis evidently saw things a little differently (and more accurately). Genesis does not specifically characterize the Old Kingdom period as a repetition of the gods (Adam I through Noah I). In fact, what we call the first five dynasties appear to be considered as an "Intermediate Period". After the Flood, there is a period in which a new dynastic line is established under Nimrod/Narmer, but it doesn't get fully set. It is only with the dynasty of Sargon (Adam II throught Noah II) that a complete new cycle is perceived. This cycle concludes with the Flood of the Middle Kingdom.
After the Middle Kingdom comes another "Intermediate Period" (of sorts) in the Genesis narrative. A second set of princes named Ham, Shem, and Japheth carry on after this Flood as the first set had done after the Flood of their day. This next transitional period ends with a killing drought and with the emergence of another Adam, known from archaeology as Akhenaten. However, the Torah ends with this "third Adam" instead of starting the history of another cycle. There apparently never was another flood in Egypt that compared to the one before the Old Kingdom and after the Middle Kingdom.
After the death of Akhenaten, there was a movement to annul his role, much as Hatshepsut was earlier suppressed. His assassinations of the Libyan pharaohs Osorkon, Takelot, and Sheshonq II probably had a lot to do with this. The Torah represents a "minority report" in which the role of Akhenaten continued to be honored. The majority opinion with the royal family was that the next Adam/Sargon would come from the "Libyan" dynasty given to Aye (Ham) rather than directly from the House of Yuya (Japheth). Ultimately, and with much more argument and bloodshed, another Adam/Sargon was recognized in the person of Piye/Sargon II.
© Charles N. Pope, US Library of Congress. All rights reserved.