The 7th day of Adar is both the birthday and death day of Moshe Rabbeinu. On the calendar, this is a very complex day. Contemporarily, we think of the 7th of Adar as a day to recognize those members of the Chevra Kadisha who care for the deceased of the community. There is a custom to fast, and when the fast is over, they are to eat a seudas mitzvah, often accompanied by a siyum (keep those ideas in mind for now). Rabbi Gavriel Zinner in his encyclopedic Nitei Gavriel lists a few interesting halachic/hashkafic considerations for Zayin Adar:
-Tachanun (everyone’s favorite): According to the Shulchan Aruch’s list of days when one cannot say Tachanun, Zayin Adar does not make the cut. Therefore, it would seem that Tachanun should be said. Furthermore, Rav Zinner quotes the Otzar HaChaim who states in the name of the Sanzer Rav, the Munkatcher Rav that it should be recited. Conversely, the minhag of the Belzer Chassidim and of the Budapest community was to recite Tachanun at Shacharis but to omit it at Mincha.
-Kiddush Levana: One is permitted to recite Kiddush Levana on Zayin Adar. The Sar Shalom of Belz, brought by the Otzar Yad HaChaim, quotes that there were those who are particular to not say Kiddush Levana until after Zayin Adar.
-Weddings: There are those who are careful to not make a wedding on Zayin Adar because of it’s status as a taanis Tzaddikim, due to it being the yahrtzeit of Moshe Rabbeinu. However, there are many who do not particularly fastidious in regard to this minhag.