Torah Reading for a Mourner During Shiva

Is a mourner allowed to be called up to the Torah, and does he need to leave during the Torah reading?

The mourner is obligated to hear the Torah reading, but he is not allowed to be called up to the Torah. On Shabbat, if he was called up, or if he is a Kohen and there is no other Kohen and he did not leave the synagogue before the reading, or if there is an aliyah he regularly receives, he goes up to the Torah, because if he does not, it would be considered public mourning.

In Shulchan Aruch, Siman 384, Se'if 2, it is written: "If the mourner is a Kohen and there is no other Kohen in the synagogue, he is forbidden to be called up to read the Torah." In Siman 135, Se'if 1, it is written: "If the mourner was called up to the Torah, he must go up, because if he refrains, it would be a public display of mourning; Rabbeinu Tam was always called up third, and when he was in mourning and the chazzan did not call him, he went up himself and said that since he was regularly called up third every Shabbat, anyone who sees that he does not go up will say it is because of mourning, and it would be a public display of mourning." The Rema writes: "And also, if the Kohen is in mourning and there is no other Kohen in the synagogue, it is permitted to call him; but in other cases, it is forbidden." The Bach and Shach in Siman 384, Se'if 3 explain that in Siman 384, where the Shulchan Aruch writes not to call him, it refers to weekdays. And in Siman 135, where the Shulchan Aruch writes to call him, it refers to Shabbat. Nevertheless, it implies that the mourner is obligated to hear the Torah reading, only he does not go up. Otherwise, the Shulchan Aruch would have written that he is not present in the synagogue. In the book Maharil (Minhagim), laws of Torah reading, it is written: "Mahar"i Segal said that a mourner is called up to the Torah on weekdays, but not on Shabbat, because then it appears as if he has a permanent standing, and he must behave in private." And although the Shulchan Aruch disagrees with him on the obligation of going up on weekdays, nevertheless, it shows that he is obligated to hear the Torah reading.