1. When did Rabbis first appear? 2. When did Bar Mitzvahs first start?

ב"ה Shalom The answer to both of your questions appears already in the Mishna of Pirkei Avot, chapters of the fathers, which is one of the earliest Rabbinic texts. Rabbis in the sense of teachers and the transmitters of the Oral Law have been around since the giving of the Torah on Mt. Sinai. In fact, in the very first Mishna of Avot, it says: Moshe received the Torah at Mt. Sinai and handed it down to Yehoshua, and Yehoshua passed it on to the elders, and the elders to the prophets, and the prophets handed it down to Men of the Great Assembly. Though the title "Moshe Rabbenu= Moshe are master and teacher appears in later Rabbinic literature, it was a role that Moshe carried out regardless of the title he carried. From that point of view, the Rabbis have always been the ultimate transmitters and interpreters of the Oral Law handing down the teachings from generation to generation uninterruptedly. If your question is when did the Rabbi become more dominant in leadership, that is a historic question, which I will also touch upon. The same above Mishna already alludes to the transition of spiritual leadership from the prophets to the Men of the Great Assembly. The Men of the Great Assembly, was a body composed of the last of the prophets , Chagai, Zecharia and Malachi and the first of the scholars of the Mishna, Shimon Hatzaddik, Simon the righteous. Once the prophets were no longer, the Rabbis filled in the gap of teaching and spiritual leadership and started becoming more dominant. This trend strengthened even more after the destruction of the Second Bet Hamikdash, when the scholars of the Yavneh era, set Judaism on its course rehabilitating the Jewish people in wake of the terrible tragedy and loss which befell the Jewish people due to the destruction. BAR MITZVA: The issue of Bar mitzva is also mentioned in the Mishna Avot, in chapter 5). Mishna 21) The Mishna says that when a boy reaches the age of 13 he is obligated to fulfill the mitzvot, the commandments. Meaning it is at this stage in life, he takes the responsibility of fully keeping the Torah and not just doing things for practice. However, in this Mishna there is no mention of a celebration for the occasion. In the 16th century in Poland ,Rabbi Shlomo Luria zt"l , instituted the Bar Mitzva as a celebration and he mentions that it already had been a tradition among Ashkenazi Jew. This is basically the source of celebrating the Bar mitzva in our time, though there are other such indications in earlier sources, such as Massechet Sofrim ( chapter 18:7) All the best