Rabbi Zalman Sorotzkin - Oznayim L'Torah - Shemos 27:20
Why was Parashas Tetzaveh chosen as the one from which Moshe‘s name would be omitted? The seventh of Adar, which is both the anniversary of Moshe‘s death and his birthday, always falls in the week of this parashah. Chazal must have wished to stress the essential purity of Israel’s faith, emphasizing that Moses the lawgiver was not transformed into the object of their worship.
It is common practice in other religions for the founder to take “center stage“ in all of their services and holidays. The birthday of the founder is their most important holiday, followed by the anniversary of his death, and fast days related to his suffering and affliction. This creates the impression that everything was great in his honor, and all their “faith“ focuses on him alone. He even glorifies himself by claiming that whoever touches the corner of his garment will have a share in the World-to-Come (as if he has control over it). And when he is no longer, his priests sell absolution from sin and heavenly portions to the highest bidder.
This is such a contrast to the ways of Jacob and his Torah! In our religion, the lawgiver humbly remains in the background. True, the learned calculated that Moshe died on the seventh of Adar and reasoned that he was born then too (since God makes the years of the righteous perfect, causing them to die on the same day they were born). But the common folk did not know Moshe‘s birthday or the day of his death, and they certainly did not celebrate these days.
Eliyahu Kitov - The Book of Our Heritage -
Customs of the Seventh of Adar (pgs. 370-371)
It is customary, in many Jewish communities, for the Chevrah Kadishah [the burial society] to observe the seventh of Adar as a day of gathering for all of its members. On this day they hold a festive banquet with the participation of the entire community. Officers of the Chevrah are chosen and regulations adopted. The day is considered as a festival for them and for the community. Why was the seventh day of Adar chosen as a holiday of the Chevrah Kadishah?
The reason for this custom reflects praise upon Israel and upon those who are engaged in the practice of Mitzvos. Men of all occupations are happy when they have a lot of work and are sad when there is little work. In the case of the members of the Chevrah Kadishah, however, although their work calls for them to perform kindness for both of the dead and the living — work which they perform faithfully — they never rejoice in their work. When are they happy? When their work ceases.
The only time when we find that the work ceased for those who perform kindness for the dead, was on the seventh of Adar, the day on which Moshe died. No mortal engaged in his burial — only God himself, and His Glory.
Another reason that the seventh of Adar was chosen as the day of gathering for those who perform kindness for the dead is that this date never falls on Shabbos.
Rav Yosef Eliyahu Henkin - Eidus L'Yisrael
List of When We Do and Do Not Say Tachanun
7 Adar is not included on this list.
Nitei Gavriel on Zayin Adar - Rabbi Gavriel Zinner
Rav Aharon Soloveitchik's Shitah - No Tachanun on 7 Adar
אִם תִּמְצֵי לוֹמַר הָכָא כֵּיוָן דְּאֶפְשָׁר לְאִיתְּשׁוֹלֵי חָיְילָא אָמַר הֲרֵינִי נְזִיר שִׁמְשׁוֹן לְאַחַר עֶשְׂרִים יוֹם וּמֵעַכְשָׁיו נָזִיר סְתָם מַהוּ הָכָא לָא אֶפְשָׁר לְאִיתְּשׁוֹלֵי מִי חָיְילָא אוֹ לָא אֲמַר כְּמֹשֶׁה בְּשִׁבְעָה בַּאֲדָר מַאי
The Gemara adds another question: If one said: I am hereby like Moses on the seventh day of Adar, what is the halakha? Is this considered an expression of naziriteship in that just as Moses passed away on that date and drank no more wine, so too, the speaker vows to be a nazirite? Alternatively, perhaps this phrase does not indicate the acceptance of naziriteship.
אמר כמשה בז' באדר מהו. מי אמרינן דהוי קבלת נזירות דמסתמא ביום שנפטר משה רבינו רבו נוזרים נזירות מחמת צער. או דלמא לא שהרי גם בז' באדר נולד וכמו כן הוא יום משתה ושמחה.
The רא״ש explains to mean that 7 Adar is either a day of צער because of Moshe’s death or a day of משתה ושמחה because of Moshe’s birth.
(א) ימים שמתענים בהם. ובו ג' סעיפים: אלו הימים שאירעו בהם צרו' לאבותינו וראוי להתענו' בהם ואע"פ שמקצתם בראש חודש יש מי שאומר שיתענו בו (וטוב שלא להשלים בראש חודש):
(ב) באחד בניסן מתו בני אהרן בעשרה בו מתה מרים ונסתלק הבאר בכ"ו בו מת יהושע בן נון בעשרה באייר מת עלי הכהן ושני בניו ונשבה ארון ה' בכ"ח בו מת שמואל הנביא בכ"ג בסיון בטלו הביכורים מלעלו' לירושלים בימי ירבעם בן נבט בכ"ה בו נהרג רשב"ג ורבי ישמעאל ור"ח סגן הכהנים בכ"ז בו נשרף רבי חנניא בן תרדיון וספר תורה עמו בא' באב מת אהרן הכהן בי"ח בו כבה נר מערבי בימי אחז בי"ז באלול מתו מוציאי דבת הארץ. בה' בתשרי מתו עשרים איש מישראל ונחבש ר' עקיבא בז' בו נגזרה גזירה על אבותינו שימותו בחרב וברעב ובדבר מפני מעשה העגל בז' במרחשוון עיורו עיני צדקיהו ושחטו בניו לעיניו בכ"ח בכסליו שרף יהויקים המגילה שכתב ברוך מפי ירמיהו בשמונה בטבת נכתבה התורה יונית בימי תלמי המלך והיה חושך בעולם שלשה ימים בט' בו לא נודע איזה היא הצרה שאירע בו בה' בשבט מתו הזקנים שהיו בימי יהושע בכ"ג בו נתקבצו כל ישראל על שבט בנימין על ענין פלגש בגבעה בז' באדר מת משה רבינו ע"ה בט' בו נחלקו בית שמאי ובית הלל:
(1) 1. The Days on Which We Fast, 3 Seifim: These are the days when troubles occurred for our forefathers, and it is worthy to fast on them. And even though some fall at the beginning of the month, there are those who say that we fast on them (and it is good not to complete a fast at the beginning of a month).
(2) 2. On the first on Nisan, the sons of Aaron died. On the tenth of it Miriam died and her well disappeared. On the twenty-sixth of it Joshua ben Nun died. On the twentieth of Iyyar, Eli the priest and two of his sons and the ark of God was captured. On the twenty sixth of it Samuel the prophet died. On the twenty-third of Sivan the offering of the first fruits in Jeruslem was cancelled during the days of Jereboam ben Nebat. On the twenty-fifth of it Rabban Shimon ben Gamliel and Rabbi Yishmael and Rabbi Hanina Deputy High Priest were killed. On the twenty-seventh of it Rabbi Hanina ben Teradion was burned to death and a Torah scroll with him. On the first of Av, Aaron the priest died. On the eighteenth of it, the eastern light was extinguished during the days of Ahaz. On the seventeenth of Elul those who slandered the land died. On the fifth of Tishri, twenty men of Israel were killed and Rabbi Akiba was imprisoned. On the seventh of it a decree was decreed against our ancestors that died by sword, hunger and plague because of the incident of the golden calf. On the seventh of Mar-Cheshvan, they blinded the eyes of Zedekiah and slaughtered his sons before his eyes. On the twenty-eighth of Kislev Jehoiakim burned the scroll that Baruch wrote dictated by Jeremiah. On the eighth of Tevet the Torah was written in Greek during the days of King Ptolemy and darkness was in the world for three days. On the ninth of it, it was not known who caused the trouble that happened on it. On the fifth of Shevat the elders who were in the days of Joshua died. On the twenty-third of it all of Israel gathered against the tribe of Benjamin on account of the concubine in Gibeah. On the seventh of Adar Moses, our Teacher, may peace be upon him, died. On the ninth of it Beit Hillel and Beit Shammai divided.
Rabbi Shmuel Marcus (Grandson of Rav Aharon Soloveitchik)
[S]ince [it] is no longer the common custom [to fast on 7 Adar,] Rav Aharon felt that nowadays it should therefore be observed as a day of Simcha which he felt was reason to omit tachanun.
Shulchanaruchharav.com/halacha/zayin-adar/
The Chabad Rabbeim did not say Tachanun on Zayin Adar starting from the Mincha of Vav Adar. Nevertheless, this custom only began once they took leadership of the movement and not beforehand. [Sefer Haminhagim [English] p. 168]
The reason: As the soul of Moshe shines within the leader of each generation, and hence they feel the value of the birth of Moshe on this day. [Likkutei Sichos 16 p. 350-351]
Rav Yecheskel Shraga Weinfeld - Table Talk - Parshas Mishpatim 5782
(ח) בז' באדר. כתב בתשו' מהרי"ל סי' ל"א המנהג להתענו' באדר ראשון וכ"כ בת"ה סי' רצ"ד אף על גב שבגמרא מוכח שאותה שנה לא היתה מעוברת וכו' ע"ש ועסי' תקס"ח, ועיין בילקוט יהושע דף ד' ע"ב דאיכא מ"ד שהיתה מעוברת ומת באדר ראשון וחד אמר שמת בז' בשבט ע"ש:
The Magen Avraham writes that in a leap year, the Minhag is to fast in Adar Rishon even if the year Moshe Rabbeinu died was not a leap year.
(Some suggest that we should apply this to all other Yahrzeit fasts to be done in Adar Rishon, while others suggest that Moshe's Yahrzeit is the exception and for all others it should be in Adar Sheni.)
This is also a way to understand why משנכנס אדר מרבים בשמחה should already start in Adar Rishon in a leap year. Being that the salvation of Purim is connected to the fact that the birth and death of Moshe Rabbeinu happened in the month of Adar, and the fast for Moshe Rabbeinu's Yahrzeit is in Adar Rishon, the joy of Adar applies to both months.
Shulchanaruchharav.com/halacha/zayin-adar/
Moshe’s birthday and Yartzite on a leap year: On a leap year, Moshe’s birthday and Yartzite are [Halachicly] commemorated on the 7th of Adar I. However the spiritual aspect behind the auspicious day applies to both 7th of Adar I and 7th of Adar II.
Such as the Simcha. This is opposed to the Halachic aspect, such as fasting, which is to be done [by those accustomed to do so] in Adar I, as ruled by the Poskim mentioned above. (Sichas 1992 2 p. 721, brought in Shaareiy Hamoadim Adar 19)