Here the word "sent" means that David sent a messenger. Alternate translation: "So David sent a messenger" (See: [[rc://en/ta/man/translate/figs-idiom]])
David was trying to find out who the woman was. The word "he" refers to David, but is a metonym for the messenger that David sent. The messenger was to ask the people for information about her. Alternate translation: "the messenger asked the people who knew her about who she was" (See: [[rc://en/ta/man/translate/figs-metonymy]] and [[rc://en/ta/man/translate/figs-explicit]])
This question gives information and can be written as a statement. Alternate translation: "This is Bathsheba ... and she is the wife of Uriah the Hittite." (See: [[rc://en/ta/man/translate/figs-rquestion]])