Jesus was not David's literal son, so this may be translated as "descendant of king David." However, "Son of David" is also a title for the Messiah, and the crowd was probably calling Jesus by this title.
Here "in the name" means "in the power" or "as a representative." Alternate translation: "in the power of the Lord" or "as the representative of the Lord" (See: [[rc://en/ta/man/translate/figs-metonymy]])
Here "highest" refers to God who rules from the highest heaven. Alternate translation: "Praise God, who is in the highest heaven" or "Praise be to God" (See: [[rc://en/ta/man/translate/figs-metonymy]])