Did Jesus believe himself to be descended from David?
No He didn't. He knew that He was, just as other people know their genealogy. If later people call it into question that is their problem. Jesus knew where He was from both in a human and a divine sense.
He did not according to Matthew 22:41-46, Mark 12:35-37 & Luke 20:41-44." If David then call him Lord, how is he his son?"
The three passages mentioned above are parallel passages which say virtually the same thing. The one from Luke is below:
42And David himself saith in the book of Psalms, The LORD said unto my Lord, Sit thou on my right hand,
43Till I make thine enemies thy footstool.
Firstly, it is important to realize that Jesus actually says nothing at all in this discussion about what He thinks or knows or believes about Himself. It is clear that He is asking His opponents a question about what is in the scriptures to bring their attention to an interesting point.
Secondly, what is interesting is the particular point which is obvious about the implications of what was said in the Old Testament scripture to which Jesus is here referring (Psalm 110:1). What is obvious is that David called his descendant 'Lord'. This is a clear reference to David's descendant Jesus Christ who is Lord of Lords and King of Kings.
Thirdly, the fact that the scribes could not Him means absolutely nothing about Jesus' own belief about Himself but it means a lot about what the scribes knew or did not know about Him and His identity. It also shows that they were ignorant of the scriptures which talked about Jesus.
Fourthly, it could be concluded from this discussion that Jesus had no problem with His own identity but He was interested to know what others thought and interested to challenge them to consider what the word of God, the God they professed to worship, also said in His word. This very word here spoke about Jesus Himself, the one to whom they were speaking.
Fifthly, in order to rightly understand scripture it is always important to read what it actually says and to read everything in context. Here the context is a discussion along the lines of a typical exchange between religious teachers or rabbis about a point of detail in scripture. The asking of questions was a regular part of such discussions.
The parallel passages from Matthew 22:41-46, Mark 12:35-37 & Luke 20:41-44 actually speak about Jesus' own self understanding that He was both descended from David and that He was one who was greater than David.
Elsewhere Jesus is referred to by a number of others who addressed Him as 'Son of David'. Nowhere does He ever correct them for saying this, another clear indication of its correctness. see Matthew 12:23, Matthew 15:22, Matthew 20:30-31, Matthew 21:9,15, Mark 10:47-48, and Luke 18:38-39.
The genealogy of Christ from King David is given in Matthew chapter 1:-
Mat 1:1 [Good News Bible] This is the list of the ancestors of Jesus Christ, a descendant of David, who was a descendant of Abraham.
Mat 1:2 From Abraham to King David, the following ancestors are listed: Abraham, Isaac, Jacob, Judah and his brothers; then Perez and Zerah (their mother was Tamar), Hezron, Ram, Amminadab, Nahshon, Salmon, Boaz (his mother was Rahab), Obed (his mother was Ruth), Jesse, and King David.
Mat 1:3-6 (SEE Mat 1:2)
Mat 1:7 From David to the time when the people of Israel were taken into exile in Babylon, the following ancestors are listed: David, Solomon (his mother was the woman who had been Uriah's wife), Rehoboam, Abijah, Asa, Jehoshaphat, Jehoram, Uzziah, Jotham, Ahaz, Hezekiah, Manasseh, Amon, Josiah, and Jehoiachin and his brothers.
Mat 1:8-11 (SEE Mat 1:7)
Mat 1:12 From the time after the exile in Babylon to the birth of Jesus, the following ancestors are listed: Jehoiachin, Shealtiel, Zerubbabel, Abiud, Eliakim, Azor, Zadok, Achim, Eliud, Eleazar, Matthan, Jacob, and Joseph, who married Mary, the mother of Jesus, who was called the Messiah.
Mat 1:13-16 (SEE Mat 1:12)
Mat 1:17 So then, there were fourteen generations from Abraham to David, and fourteen from David to the exile in Babylon, and fourteen from then to the birth of the Messiah.
The Jews took their genealogies very, very seriously; when Christ said He was descended from David you could absolutely guarantee that the Pharisees would have checked out the genealogy of Christ very closely, and if there had been anything wrong they would have trumpeted it to the world. In fact, His genealogy would have already been checked and then remembered from soon after His birth when the wise men from the east came to Herod seeking Him. Herod was rightly terrified: he was a usurper king who was not even entitled to sit on the throne of David because he was not one of his descendants, he was not even Jewish (he was Idumaean), and here was Someone fully entitled to claim Herod's throne because of birth!
Mat 2:1-3 MSG After Jesus was born in Bethlehem village, Judah territory--this was during Herod's kingship--a band of scholars arrived in Jerusalem from the East. [v. 2] They asked around, "Where can we find and pay homage to the newborn King of the Jews? We observed a star in the eastern sky that signaled his birth. We're on pilgrimage to worship him." [v. 3] When word of their inquiry got to Herod, he was terrified--and not Herod alone, but most of Jerusalem as well.
The Pharisees never said Christ was not entitled to call Himself King: they knew it and recognized it , and put a sign saying He was king on the cross when He was crucified.
Mat 27:37-38 KJV And set up over his head his accusation written, THIS IS JESUS THE KING OF THE JEWS. [v. 38] Then were there two thieves crucified with him, one on the right hand, and another on the left.