God's Promise

According to what "promise" (Acts 13:23) did God promise that Jesus would be from David's seed?
According to the one conveyed by the prophet Isaiah - "For unto us a Child is born, unto us a Son is given; and the government will be upon His shoulder. And His name will be called Wonderful, Counselor, Mighty God, Everlasting Father, Prince of Peace. Of the increase of His government and peace there will be no end, upon the throne of David and over His kingdom, to order it and establish it with judgment and justice from that time forward, even forever. The zeal of the Lord of hosts will perform this." (Isaiah 9:6-7) - as well as by archangel Gabriel: "Now in the sixth month the angel Gabriel was sent by God to a city of Galilee named Nazareth, to a virgin betrothed to a man whose name was Joseph, of the house of David. The virgin’s name was Mary. And having come in, the angel said to her, “Rejoice, highly favored one, the Lord is with you; blessed are you among women!” But when she saw him, she was troubled at his saying, and considered what manner of greeting this was. Then the angel said to her, “Do not be afraid, Mary, for you have found favor with God. And behold, you will conceive in your womb and bring forth a Son, and shall call His name Jesus. He will be great, and will be called the Son of the Highest; and the Lord God will give Him the throne of His father David. And He will reign over the house of Jacob forever, and of His kingdom there will be no end." (Luke 1:26-33)

How did God keep His promise?
Through the following lineage: "Now Jesus Himself began His ministry at about thirty years of age, being (as was supposed) the son of Joseph, the son of Heli, the son of Matthat, the son of Levi, the son of Melchi, the son of Janna, the son of Joseph, the son of Mattathiah, the son of Amos, the son of Nahum, the son of Esli, the son of Naggai, the son of Maath, the son of Mattathiah, the son of Semei, the son of Joseph, the son of Judah, the son of Joannas, the son of Rhesa, the son of Zerubbabel, the son of Shealtiel, the son of Neri, the son of Melchi, the son of Addi, the son of Cosam, the son of Elmodam, the son of Er, the son of Jose, the son of Eliezer, the son of Jorim, the son of Matthat, the son of Levi, the son of Simeon, the son of Judah, the son of Joseph, the son of Jonan, the son of Eliakim, the son of Melea, the son of Menan, the son of Mattathah, the son of Nathan, the son of David." (Luke 3:23-31)

Why does Paul specify that "John had first preached, before His coming" (Acts 13:24)?
John "preached... the baptism of repentance" (Acts 13:24) - the need for people to recognize their sinfulness and seek God's help to turn away from and to seek a solution to their sins. The recognition of the problem - sin - had to precede its solution - Christ - back then, and it must precede the solution today. The tendency of pulpits today to preach the cross of Christ but to skim over the issue of sin amounts to offering the solution without explaining what it solves.

What did John the Baptist mean about the "sandals" (Acts 13:25)?
See John the Baptist.

How was the "word of this salvation" (Acts 13:26) sent to the Jews and God-fearing gentiles in Antioch of Pisidia?
By Barnabas and Paul, who is speaking.