The "Sonof" Begats in Luke
First of all, why is there a genealogy at this point? Who rates genealogies in the Bible? Patriarchs and royalty, David for example.
The discrepancy with Matthew's genealogy could mean: 1) That Matthew made changes to fit things into his scheme of "14 generations from Abraham to David, 14 from David until the exile, and 14 from then until the Messiah." Or 2) Both are fictional.
From NT Wright's The New Testament and the People of God, (pg 319) re messianic expectations in the first century:
"1. Expectation was focused primarily on the nation, not on any particular individual. The hope [of deliverance from pagan domination & divine displeasure]... remains fundamental, occurring far more widely than expressions of hope for a Messiah or similar figure. Sometimes, indeed, texts are might be thought to speak of a Messiah are referred to the whole community, a process which is already visible within the Hebrew Bible itself.
"2. This expectation could, under certain circumstances, become focused upon a particular individual, either expected imminently or actually present. The circumstances under which this was possible seem to have been threefold; the appearance of an opportunity (such as at the death of Herod), the particular pressure of anti-Jewish action by pagans (such as under Hadrian), and the crescendo of speculation connected with the attempt to work out messianic chronology.
"3. When this happened, the generalized expectation of a coming figure can be redrawn in a wide variety of ways to fit the situation or person concerned. Davidic descent can clearly be waived. The idea of two Messiahs is not a contradiction in terms. The particular felt needs of the time can influence the presentation: Herod could hope for his son to be the true king; the Sicarii could put forward Menahem, or the peasants Simon bar Giora...."
"Davidic descent can clearly be waived."-- And in all three synoptic gospels, we find Jesus himself arguing against the belief that the Messiah must be descended from David. In the case of David himself, the crucial fact was not his pedigree, but his anointment by Samuel... and his prestige among the people.
Another detail that also comes up in the synoptic accounts of Jesus arguing on the Temple grounds, when the authorities ask him: Where does he get his authority? He asks them whether they accept John the Baptist's authority. It's a minority view, but clear to me personally, that Jesus became officially Messiah (de jure king of Israel) from his anointing by John the Baptist, that this was why the gospel writers all mention his encounter with John. If so, "Baptism" (though it might be ceremonially appropriate for a man who could be taken to "represent" Israel in some sense) was not the significant point of their meeting.
There is, as it happens, some question about Jesus' paternity. I myself wonder about the medieval Jewish legend that he was fathered by a Roman soldier (There was, after all, a punitive raid through Galilee near the time of his birth.) That legend has not been traced back to any source near the first century; but it is odd that his fellow villagers are said to refer to him as "the son of Mary", which was not a nice thing to call someone in that culture. It could be a misquote by gospel writers who didn't think he could be a "son of God" if he was simply the son of Joseph. And how he physically came to be born... might just have very little to do with who he was, and is.