Are Dati Leumi Jews considered Yeshiva/Litvish, Haredim, or Hassidic? Are there overlaps? How do you know? What does Yeshiva/Litvish mean and how does this group differ from other groups? Do they also follow Rabbi Kook? Thank you.
In general, I try to stay away from labels, especially when these terms can change from place to place. But for clarification sake, Dati-Leumi means religious-Zionist or "nationalist", and can and historically does have representatives in all streams of orthodoxy. Although the term "Haredi" means one who is in awe of God, and originally referred to all religious Jews, today, it is usually used regarding the Hassidic and Yeshivish/Litvish (=non-hassidic haredim), as well as the Sefaradic "Shas" element. The learned in these groups all admire and respect Rav Kook, yet generally don't study his writings, for the usual yeshiva curriculum is to learn just the Talmud, aside from the fact that Rav Kook's philosophy is very difficult to understand. In addition, Rav Kook's writings are innovative, something often feared in those conservative circles. The aforementioned haredim tend not to serve in the army, and generally see Judaism identically to what was practiced throughout the 2,000 year exile, as opposed to Rav Kook who calls to revive the original Torah as found in the Tanach: living as the Nation of Israel, in the Land of Israel, speaking the Holy Language and serving in the Israeli army. Today, there are more and more overlaps, especially among the "Hardal"= Haredi-Leumi, who often study chassidut, and dress just as modestly and learn as much Torah as the haredim.