I don't have very good answers for your questions. I am perhaps being imprecise in my recollection of exactly what terminology is used by Metropolitan Hierotheos (Vlachos) in his writings on the teachings of St. Gregory Palamas regarding theosis and the Theotokos.
The liturgical texts say that Christ "took flesh of the Virgin." But on the other hand, once he was conceived, he was a unique person, so it only makes sense to say that his body contains his own flesh and blood. The emphasis of St. Gregory Palamas is on the intimate association of Christ's body and blood with that of the Theotokos during his 9th months in her womb, as I recall. Again, my recollection and wording may be imprecise.
I hadn't really thought about the question of why Christ "needed" a mother. I suppose that I always thought that God expected mankind to work its way back to him. This began outside the gates of Eden, where Eve is told that her seed would bruise the serpent's head. Also, while it isn't often talked about as such, the entire human race shares a common nature. We are all linked physically to each other through our common ancestry in Adam and Eve. For Christ to fully share our nature, he would need to be physically linked with us in that common ancestry.
Part of the emphasis in the two lineages given in the Gospels of Matthew and Luke is the role of adoption. Christ was the son of Joseph by human adoption, and likewise Joseph himself was of a kingly line (St. Matthew's lineage) through adoption (several fathers give this explanation.)
The lineage in Luke is also a geneology leading to Joseph, again showing that he was of the seed of David, but it is a strict blood-line, with no adoption involved (adoption being in the form of a man "raising up seed" to his brother dies without an heir. The fathers explain that the two lineages show that whether one "counts" these "raising up of seed" for a brother or not, Christ is descended from David. And in either case, at the final point in the lineage, Christ's lineage is always traced to his adoptive father, Joseph, in the Gospels. It is a male lineage.
This is appropriate, since we are children of God by adoption -- not by nature or essence.
There is no geneology anywhere in Scripture of the Theotokos herself. It is my understanding that she was of the tribe of Levi (thus her cousin Elizabeth being married to a high priest.) Others, based on the fact that Joseph was an elderly distant relative of hers, believe that hers was either also of the tribe of Judah and general lineage of David, or that her family was of both tribes.
This would be not at all unlikely, since the "lost" ten tribes of Israel included the tribe of Levi. The Jewish nation at the time of Christ was primarily of the tribe of Judah, but doubtless had remnants of all of the other tribes absorbed into it. Some traditions have claimed that the 12 apostles came from lineages of the 12 tribes.
Either way, in Christ, we have the union of the priestly and kingly lineages within the nation of Israel.
I'm not sure about your questions about Christ's genetic likeness to us. Are you raising the question of whether he was haploid or something? :-)
At the risk of sounding blasphamous, I would say that He would be genetically a woman if His was the flesh of Theotokos. And, technically speaking, there was no conjunction of genetic material; He would have XX make-up, a Barr body, and no Y chromosome. I am not questioning if that is possible -- I am saying that for Him to be "fully man" as we are in every sense but the pre-Fall state does not agree with our understanding of humanity biologically, as it is implied when we say "in every way human" like us.
FWIW, I've been taught that the Luke geneology is actually through Mary, not Joseph. The argument goes that if Joseph had been the blood father of Jesus, then Jesus would be barred from the throne according to the Matthew geneology. This is because it includes the name of (Jehoiachin) Jeconiah (v. 1:11). Apparently, there was a curse on this Royal line:
Jer. 22:30 : This is what the LORD says: "Record this man as if childless, a man who will not prosper in his lifetime, for none of his offspring will prosper, none will sit on the throne of David or rule anymore in Judah."
So, this would not be the best line to use because most people would assume Joseph was the blood father. Instead, He uses the Luke geneology, which traces back through Nathan, not Jeconiah. This is Mary's line. Now, I saw in your post the requirement that all blood lines have to be traced through the male. One argument is that there is an exception spelled out in the Bible. Here is an excerpt from an article I found on the subject: The Lineage Loophole
"However, many of the people that teach on the genealogies fail to realize or address a major problem associated with the genealogical listing found in Luke's gospel, the lineage of Mary. Once you have established that the line is indeed Mary's you must deal with a second difficulty. The rights of the line are not passed through the mother, only the father. Even though Mary, through her lineage, was of the Davidic bloodline, she should be excluded from being able to pass those rights of the bloodline because of being a female (Deut 21:16). So it is not enough to prove that Mary was an unblemished descendant of David, she had to be a male to transfer the rights. Therefore she would be disqualified to transfer the rights to her son Jesus, except for a little known exception to the rule."
"In Numbers 26 we are introduced to Zelophehad. Zelophehad, we are told, had no sons, only daughters. In Numbers 27, following the death of Zelophehad, the daughters of Zelophehad came before Moses and argued their plight. Because their father had died with no sons, all of their rights of inheritance were to be lost and they felt this was unfair. So Moses prayed to God and God gave Moses an exception to the rule. The Lord told Moses that the inheritance CAN flow through a female, IF they fulfill two requirements. There must be no male offspring in the family (Num 27:8) and if the female offspring should marry, they must marry within their own tribe (Num 36:6)."
Anyway, it's just another view. I thought you all might be interested. :)