Q. Did Rav Kook have a beard before he was married? Did he want his students to grow long Payos (long earlocks)?

A. Although every detail in Torah is important, the customs which you are referring to are clearly not major points nor even mitzvot in the Torah, nor in Rav Kook's outlook on the world. I suggest spending more time on his beautiful, original and spiritually-awesome philosophy and understanding of Torah, life, morality & spirituality, which are much more basic and central to his (and to every rabbi's!) Judaism, rather than his external garb. Nevertheless, Rav Kook does discuss the significance of a beard and payot (Otzrot HaRe'iya, p. 717-719) explaining the benefit in having a special, national "look" which identifies us as the Nation of Israel, even regarding our physical hair grooming. People tend to find beauty and love among those who fit their national appearance, for example, Japanese usually see beauty in the way they themselves look, so it adds to that all-important love of Israel. Together with the importance and beauty of individuality, Rav Kook stresses that National Unity and pride are also important. Especially Israel, Hashem's Chosen Nation, should patriotically stand out, identify and be identified, with our unique role. Additionally (ibid, 912), he writes that one is definitely permitted to shave (without a razor-blade), yet it is commendable not to do so, for the aforementioned reason of national pride. Accordingly, Rav Kook himself probably grew a beard even before his wedding, following the aforementioned ideas and midat chassidut (although there are no known photos before age 23, when he was already married and serving in the rabbinate), and yes, Rav Kook grew payos. But more important is we find in the pictures of his yeshiva students already in Rav Kook's time, almost none dressed as the way Rav Kook was raised and dressed in Lithuania, but contrarily were well-groomed in the modern fashion, the vast majority without beards and almost none with visible payos. This is in line with what the Rav himself writes that it's important that yeshiva boys dress modernly in a way that is seen as relevant to the generation at-large, "the living movement of the [chalutzim/Zionist pioneers], with [their] joy of life and bravery, intellectual broadness, and national pride... cannot bear the hunched-back.. sadness.. which transmit sadness, dislike for [the world].. the [old-fashion] eastern clothing when mixed with poverty awakens disgust for those accustomed to [modern] dress" (Igrot HaReiya 146). Similarly none of the students of Rav Zvi Yehuda, his loyal son and successor, grew payos while in his yeshiva, for Merkaz HaRav Kook students, who were afterwards the Religious-Zionist rabbinic leaders, were taught to always focus on the internal spiritual growth and not external trappings which can often even mislead and distract from the essence [Rav Zvi Yehuda was very careful of the laws of "Yuhara", not doing public acts which express one's piety in a locale where that action is not generally accepted unless referring to real mitzvot (not customs), e.g. he often mentioned the obligation not to hide one's tzitzit (halachic, for Ashkenazim), on the other hand, Rav Yochanan Fried and Rav Aviner told me that he didn't lay Tefilin Rabbenu Tam.] In short, regarding a "look" which doesn't have halachic basis, Rav Kook believed his yeshiva boys and bnei Torah should dress in a way conducive to bringing the masses to love the Torah and see it as a living modern and relevant ideal, but this differs depending on time and place. In other words, nobody knows what Rav Kook would have said today in a Jewish State were it not uncommon to grow payos, but they surely should be orderly & well-kept.