It is forbidden to leave the synagogue from the beginning of the Torah reading until its conclusion. Even someone who already heard the Torah reading is forbidden to leave. If one does leave, he offends the respect of the Torah. Of him it is written (Isaiah 1:28), “Those who abandon God will perish.” One who must leave, such as a person whose only ride to work is about to depart, is permitted to leave between aliyot, for then the Torah scroll is closed and the offense to the respect of the Torah is minimal (Shulchan Aruch, Orach Chaim 146:1).
In principle, there is no obligation to stand throughout the Torah reading, though there are some who enhance the mitzvah by standing, just as all the Jews stood at Mount Sinai (Rama 46:7). Some are strict to stand when answering “Baruch Hashem HaMevorach L’Olam Va’ed” since it is a matter of sanctity, and the recital of a matter of sanctity requires that one stands (Mishnah Berurah 146:18). According to many, it is unnecessary to stand when the Torah is being read, and that is how the Shulchan Aruch rules (146:4). The Ari HaKadosh was also accustomed to sit throughout the whole reading, even when Barchu was recited (Kaf HaChaim 146:20). Indeed this is the custom in many Sephardic and Ashkenazic synagogues.
From the time the Torah scroll is opened to be read, the people in the congregation are forbidden to talk to one another, even concerning matters of halachah (Sotah 39a). There is an opinion which permits a few brief words of Torah between the aliyot, provided that the exchange does not continue into the berachot or Torah reading (Bach). Some say that it is even forbidden to talk matters of Torah in-between aliyot and that it is only permitted to learn alone at that time (Eliyah Rabbah). It is good to be strict concerning this, because if people start to talk about matters of Torah, it will be hard to stop when the person called up begins to recite the blessing.
It is permissible for a rabbi to answer an urgent question in the breaks between the aliyot (Mishnah Berurah 146:6). Gabbaim are also permitted, during the breaks, to talk about essential matters that are pertinent to the prayer service. When there is no other option, it is permitted for a rabbi to answer questions even at the time of the Torah reading. Likewise, the gabbaim are permitted to talk about pressing issues which demand immediate attention; for instance, how to avoid insulting one of the people praying who expects to be given an aliyah.