1. The Radak on Tehillim (81:4) ונראה כי זה הוא טעם מצות התרועה בראש השנה לפי שבטלו אבותינו מעבודתם ונחו מיגיעם ביום זה והתרועה הוא סימן שלוח עבדים וצאתם מרשות אדניהם וכן צוה הקב״ה תקיעת שופר ביום הכפורים של יובל שהיה סימן לשילוח עבדים וצאתם מרשות אדוניהם כן צוה הקב״ה לתקוע בראש השנה לזכר כי ביום זה בטלה העבדה והיה תחלת חירותנו. According to Radak we blow shofar to remind ourself that it is the day the servitude in Egypt ceased.
2. Midrash Emor (29:1) בשנים עשר יצא בדימוס אמר הקדוש ברוך הוא לאדם זה סימן לבניך כשם שעמדת לפני בדין היום הזה ויצאת בדימוס כך עתידין בניך לעמוד לפני בדין ביום זה ויוצאין לפני בדימוס אימתי "בחדש השביעי באחד לחדש. How was Adam vindicated he was cursed and thrown out of Gan Eden? Furthermore, not everyone comes out of the din vindicated?
The Sharay Hamoadim says Chazal mean that the din is an opportunity for tikkun. Without the din on Adam he would have been stuck with his sins, the judgement allowed him the opportunity to obtain a tikkun for himself. So too, the judgement on Rosh Hashana allows us to refocus to find our proper path.
The Tur says that we prepare as a festival for RH since we know that Hashem will do for us a נס and we will come out vindicated. Why does he say it is a נס, why does he not say we are confident in emerging righteous in the judgement? It may be that it requires a נס to emerge indicated in judgement. In light of the above idea, one can translate נס as raising up as in נתת ליראיך נס להתנוסס for the judgement is a way to elevate ourselves.
3. The Ran cites the opinion of the Geonim that when Shabbos is Rosh Hashana since we do not say hallel we don't say צדקתך. What is the connection? I would presume the Geonim are of the opinion that we don't say צדקתך only when there is hallel said. The Ran cites those that disagree for it is no worse than any other Rosh Chodesh. But there is no hallel on this Rosh Chodesh? Presumably they hold like the practice that any day without tachnun has no צדקתך. This is the custom of the Shulchan Aruch yet he still rules to say צדקתך on Rosh Hashana? The Besi Yosef has another severah that since it is a day of judgement it makes sense to say צידוק הדין on Rosh Hashana.
4. Tosfos (32a) asks why why we say the pessukim from כתובים before נביאם in the מלכיות זכרורת שופרות? The pessukim from נביאים that we recite discuss the future days and hence we put them at the end.
5. Tosfos (16b) cites Beha"g that if a tzibbur has an אונס and can't blow shofar it is considered a year that shofar was not blown but that does not apply on Shabbos. why is Shabbos different from an אונס? See Meshech Chachma Emor. According to the explanation of Chassidus that on Shabbos the idea of shofar is accomplished even without the actual shofar it is different from a regular ones where you are still missing the shofar.
6. The Ramban (cited in Ritva RH 26 and in his derasha on Rosh Hashana) says that a shofar by definition is the horn detached from the animals after the marrow bone is removed. All horns in their attached stage are called קרן and are called shofar when detached from the animal and fit for blowing. What emerges from the Ramban is that a shofar is something totally detached and removed from the live animal. The lesson here is that the the shofar must be removed from the physical being. This may be why the shofar blowing is the symbol of freedom for the Maharal explains freedom as being removed from physical boundaries (Z'eav Yitraf Ch. 35.)
7. The Gra says that the Hallel of Rosh Hashana and Yom Kippur is made up on Chanukah. We see that RH and YK are not excluded from Hallel but the Hallel in moved to Chanukah. In other words, on RH and YK it is impossible to recite the Hallel because of ספרי חיים ומתים פתוחין but there is an obligation of Hallel. How does Chanukah serve as a makeup time? Chanukah celebrates out victory over יון. Chazal equate יון with חושך. The opposite is RH which is called אורי. This is because on RH we focus on the clarity of Hashem's kingship and dominion over everything. יון attempted to take G-d out of the picture of the world and the victory of Chanukah attested to a return of the light. Hence, Chanukah is the makeup for the Hallah of the Yomin Noraim.
9. In a letter, the Rebbe's father points out a connection between the three woman who were נפקדה on RH, שרה רחל חנה and the unique letter of each name, ש, ל, נ being the same as the dimensions of the תיבה. He points to a Zohar to make a connection between the תיבה and RH. It is noteworthy as the Berditchever points out that these letters spell the word לשן, speech. These great woman used their power of speech to cry to Hashem to merit children.