Why did he say "v'Atah"?
Malbim: Also this is a consequence of the faithful Chasadim of David. He is unlike others, who can ask from Hashem only according to the preparation, for Hashem puts influence on a person only according to his preparation - Hashem gives Chachmah only to one who has Chachmah (Brachos 55a, from Daniel 2:21 or Shemos 31:6). That refers to what depends on a person's own merit. The kingship based on Chasdei David is different. You need not look at my preparation. Rather, You will bestow on me Chachmah so that the Bris of Chesed that You made with David will not be annulled
How old was Shlomo at the time?
Rashi: He was 12 years old. He was born when Amnon raped Tamar - "va'Yhi Acharei Chen ul'Avshalom Achos?" (Shmuel II, 12:25). After two years, Avshalom was shearing his flock and killed Amnon. He fled to Geshor for three years, and was in Yerushalayim for two years, and then rebelled; Shlomo was seven at the time. After there were three years of famine, the next year David counted Yisrael; it took nine months. In the year that David died, he enacted Mishmaros 1 - "bi'Shnas Arba'im l'Malchus David Nidreshu" (Divrei ha'Yamim I, 26:31); Shlomo was 12 when David died.
What is the source that this was the year after he counted Yisrael? (PF)
Why did he say "Atah Himlachta Es Avdecha Tachas David Avi"?
Malbim: My kingship is due to You; it is not due to me and my merits. Even so, You will bestow upon me according to Your Chesed and Your promise to David. Now, I cannot conduct kingship, for I am a small youth (I will need many years to learn Chachmah).
What is the meaning of "Tzeis u'Vo"?
Radak: This is going in front of the nation, like "Asher Yavo Lifneihem va'Asher Yetzei Lifneihem" (Bamidbar 27:17).