Why does the Sefer mention David's children now?
Radak, Malbim: Above (2:10-15), it mentioned Ram's family, until David. It did not list David's children there, in order to give Kalev's lineage. After it finished, it returns to complete David's lineage.
In Shmuel II, 3:3, it says that his second son was Kil'av!
'Rashi', Rashi (Shmuel II, 3:3), Radak and Malbim, from Midrash va'Ychulu and Tanchuma (Toldos 6): Scoffers of the generation said that Avigayil was already pregnant from Naval (with Daniel) when David married her. 1 Therefore, Hashem made Daniel Kulo Av 2 (so he is called there Kil'av).
Rashi citing Berachos 4a: He is called Kil'av because he was HiChLim (shamed) Mefiboshes in Halachah -- "Chacham Beni v'Samach Libi v'Ashivah Chorfi Davar."
Radak: He had two names. There are many people like this.
Malbim: We must say that Havchanah (waiting three months before marrying a widow or divorcee, to avoid all doubts about lineage of her children) is only mid'Rabanan, and it was not yet enacted. David married her after her [month of] Aveilus. (Perhaps he relied on the majority of births, which are in the ninth month of pregnancy, so it suffices to wait one month! ? PF)
Radak: I.e. totally like his father's form, to remove suspicion. He called him Daniel, for Danani Kel mi'Naval (Hashem judged between us, and vindicated me ? PF). Malbim ? people were calling him Kalev, for Naval was Kalbi (from Kalev's family).
Why is Avigayil called "ha'Karmelis" here, and in Shmuel II, 3:3 "Eshes Naval"?
Malbim (Shmuel II, 3:3): There she is called Eshes Naval, for it says there that her son was called Kil'av, because she was previously married to Naval (refer to 3:1:2:1).
