Is the wife obligated to walk behind her husband, based on the story of our master David (peace be upon him) and the Sheikh's daughters, and is there a Hadith that supports this?

There is no Shar'i (Islamic legal) impediment that prevents a woman from walking beside her husband. Rather, the story of the Prophet, peace and blessings be upon him, racing with Aisha, may Allah be pleased with her, confirms the permissibility of this. As for the story of Musa (Moses), peace be upon him, with one of the Sheikh's daughters, there is no proof in it (to prohibit a wife walking beside her husband) because he was not her husband, and his walking behind her might have exposed him to looking at what is impermissible.