What is the meaning of "va'Yeshev
Rashi: He sat in front of the Aron.
Radak citing Yuma 69b: This hints that only kings from Beis David may sit in the Mikdash.
Radak (from Shochar Tov 1, Yerushalmi Pesachim 5:10): Even kings from Beis David may not sit in the Mikdash. Above [angels] do not sit - "Serafim Omdim mi'Ma'al Lo" (Yeshayah 6:2), "v'Chol Tzeva ha'Shamayim Omdim Alav" (Melachim I, 22:19), "Bein ha'Omdim ha'Eleh" (Zecharyah 3:7) - may David sit?! Rather, some say that he supported himself [on the wall] 1 , and some say that he settled himself in prayer 2 .
Pnei Moshe (5:10): This is like sitting. (Therefore it says va'Yeshev. It is not totally like sitting to be forbidden. Our text of the Yerushalmi implies that this answer was rejected; also Shochar Tov connotes like this. - PF)
Malbim: Since he was told that his kingship will be forever, he thanked for this, and prayed for the future.
Why did he say "Mi Anochi"?
Radak: Am I worthy of kingship?! 1
Radak citing Shochar Tov 1: I come from Rus ha'Mo'aviyah. Not only did I enter Klal Yisrael. You even made me king!
Alshich: I (and all others) do not deserve to be here after the sin of Adam ha'Rishon. We exist only because You are "Hashem Elokim" - merciful in judgment.
Chomas Anach: There was doubt about David's conception, for Yishai intended to have Bi'ah with his Shifchah [in order to be Metaher his seed 2 (Yosef Tehilos 69:9, citing the Rama mi'Pi'ano); like the method given in Kidushin 69a].
Why did he say "u'Mi Veisi"?
Radak #1, Malbim: Is my family worthy of kingship?! 1
Radak #2: Are my sons worthy of kingship?!
Alshich, Metzudas David: My family comes from Rus ha'Mo'aviyah. Not only did I enter Klal Yisrael. You even made me king!
Even if David did not consider himself worthy, why did he say so about his family? Yishai taught to 600,000, and was one of four who never sinned! - PF)
What is the meaning of "Ad Halom"?
Rashi: You made me king. Radak citing Shochar Tov 1 - we learn from "Hava Od Halom Ish