49. The Trisker Magid (Magen Avraham, beginning of Vayikra) writes that when Esther became queen, the king sent gifts to his subjects (see Esther 2:18, עשה למדינות והנחה). This is the custom of kings and queens; when they are crowned, they bestow gifts upon their subjects. So, on Rosh Hashanah, when keviyachol, we have the privilege to crown Hashem as King over the world, Hashem bestows His many brachos on us.
The Gemara (Rosh Hashanah 11.) states that the slavery in Mitzrayim ended on Rosh Hashanah. Similarly, each person can be released from his problems on Rosh Hashanah.
It states (Tehillim 33:22) לך יחלנו כאשר עלינו 'ה חסדך יהי. Reb Moshe Leib Sassover zt'l explains: We ask Hashem to bestow His kindness upon us, even though He sees that we are unworthy. The reason is לך יחלנו כאשר; we trust in Hashem although we never saw Him. So, too, we request that Hashem should bestow His kindness on us, even when He doesn't see us as being worthy of His kindness.
We say in a piyut on Rosh Hashanah, וישר צדיק אמת פעלתו אמונה עצתו יושר סתרו נקמה נאפד. Rebbe Michel of Zlotchev zt'l explained: Sometimes a person feels, נקמה נאפד, that Hashem is acting with anger and that Hashem is against him. יושר סתרו, Hashem's attribute of יושר, correctness, סתרו is concealed from him because he doesn’t understand why he deserves the punishments he is going through. אמונה עצתו, the counsel is to have emunah that this is Hashem's will. אמת פעולתו and then he will merit to understand that Hashem is acting towards him with the truth. וישר צדיק, and he will know that Hashem’s ways are righteous and correct.
Throughout the year, the chazzan begins the tefillah, standing at the amud. On Rosh Hashanah, the chazan begins המלך from his seat, wherever he davens in the beis medresh, and only then does he go over to the amud. The reason for this custom is unknown (see Nitei Gavriel).
Perhaps it is to remind people that wherever they are, whatever tests and challenges they go through, they are in the most ideal place to announce Hashem as King from there.
It states (Tehillim 17:2), יצא משפטי מלפניך. The Damesek Eliezer zt'l explains, Hashem will make the judgment לפניך, as you want it. Whatever you want for the new year – health, wealth, whatever you desire – it will be yours, as you desire it. The only condition is, מישרים תחזינה עיניך, that you look at everything through the eyes of emunah.
It states (Yirmiyahu 31:19) אזכרנו זכור בו דברי מדי כי 'ה נאם ארחמנו רחם לו מעי המו כן על עוד, "'Whenever I speak of him, I remember him more and more. Therefore, my inner self yearns for him, and I will surely take pity on him' – says Hashem." and we say this pasuk in the Malchiyus. Rebbe Mordechai of Lechovitz zt'l explains בו דברי מדי כי, I speak to myself, and I think about how often I remember Hashem, and I ask myself, זכור, "Is that called remembering Hashem? It isn't enough! אזכרנו עוד, "I must remember Hashem more!" מעי המו לו, this arouses Hashem's compassion on him. Hashem will say, רחם, "Is my compassion enough? עוד ארחמנו, "I will have even more compassion on him."
