2) The gemara in a number of places writes that an aveil is chayav in all mitzvos except for tefillin (on the first day of aveilus). All mitzvos? Isn’t an aveil not allowed to engage in talmud Torah, asks the Ritva in Moed Katan (15)? The Ritva gives two answers: 1) an aveil is exempt from talmud Torah and the gemara was just giving a general rule to which there are exceptions; 2) an aveil is chayav in talmud Torah and fulfills that obligation with the minimum of reciting shema. According to the first answer, it seems that learning topics like aveilus and midrashim about the churban, which according to the Ritva are permissible both during aveilus and on 9 Av, is not done for the sake of fulfilling the chiyuv of talmud Torah – no such chiyuv exists. Rather, these limudim are permissible as part and parcel of absorbing the message of the day and engaging in aveilus.
Shu”T Divrei Yatziv (O.C. 240 here) writes that the reason we abstain from learning on 9 Av is to show that our mourning is not primarily about our loss of sovereignty – it’s about the bitul Torah caused by galus (see Chagigah 5b). We ask Hashem to bring geulah so that the glory of Torah should be returned. (I think you can say the same vort even if you don’t agree that bitul Torah is the primary focus of our mourning.)
3) It’s not clear why the shir shel yom is pushed from shacharis to minchah on 9 Av. Since you are allowed to say everything that is part of the normal seder of tefilah, why is the shir shel yom any different than, for example, korbanos?
4) The Navi promises that the fasts we have for the churban will one day become celebrations. Tisha b’Av itself is called a “moed” and we skip tachanun (according to some views ironically an aveil may not daven for the amud on 9 Av because it is a moed). B’shlama cancelling the aveilus and fasting, we understand that once there is a complete geulah there is no need to mourn, but why must these days gufa become transformed into yamim tovim?
Hashem does not engage in destruction for the sake of destruction – every soseir is soseir al menas livnos. What the Navi is telling us is that these difficult days themselves, which to us appear so filled with suffering, are in truth the building blocks of geulah. The very same events which we look at as tragic today will one day become sources of celebration.