Why does it say that David ruled over all of Yisrael?
Radak: He won all the wars, and Beis Sha'ul was wiped out, so no one had any doubt about his kingship.
Why does it say that David did Mishpat and Tzedakah?
Rashi (from Sanhedrin 49a): Because David did Mishpat and Tzedakah, Yo'av succeeded in war (verse 16 hints to this). Yo'av enabled David to do Mishpat and Tzedakah. David judged, and Yo'av enforced his rulings. Also, because Yo'av engaged in wars, David was not distracted with them, and his heart was open to judge correctly.
Targum Yonasan: He did correct judgment and merit. Radak - he would judge the case between the litigants, and do Tzedakah when they needed. R. Yehudah (Devarim Rabah 5:3) - he vindicated the innocent and obligated the liable, and if the liable could not pay, David paid for him.
Kli Yakar citing R. Nechemyah (Devarim Rabah 1 , Shoftim 3): He was Matzdik (justified) the Gazlan's soul via removing the theft from his hand.
Malbim: They requested two matters from the king (1:9:20) - "our king will judge us
Devarim Rabah (5:3, in the standard (Vilna) edition): He vindicated the innocent and obligated the liable - this is Mishpat. It was Tzedakah to take the theft from the thief's hand.
