As a rule, the Amida is supposed to be recited quietly (Peninei Halakha: Prayer 17:7). However, on Rosh Ha-shana and Yom Kippur, there were those who recited the Amida out loud in order to improve their concentration. Even though it is not permissible during the rest of the year, as it may confuse other worshippers, this is not a concern on Rosh Ha-shana and Yom Kippur because everyone has a maḥzor (SA 582:9). Nevertheless, people should not pray too loudly, both to avoid disturbing other worshippers and to avoid appearing like the prophets of Ba’al, who screamed at their gods (Rema 101:3; MB ad loc. 12).
Another reason to recite the Amida quietly is its exalted nature, which dictates that it be kept private (MA OḤ 101:4; MB ad loc. 11). On the Days of Awe, it makes sense to be even more careful about this. Where almost everyone prays silently, as is the case in most congregations today, one may not pray out loud. Even though there is no concern that people will make mistakes, as they have maḥzorim before them, nevertheless, when one prays aloud, it distracts worshippers and disrupts their concentration.
As maḥzorim note, it is customary to open the ark during the recitation of certain prayers. At those times, it is customary for everyone to stand in order to give honor to the Torah, which is on display. However, according to the letter of the law, the obligation to stand is limited to the time when the Torah scroll is in motion. When it is stationary, whether in the ark or on the podium, one is not required to stand. Therefore, the elderly, the weak, and the sick, who find it difficult to stand up, may sit even while the ark is open. Nevertheless, when the Torah is in motion, they should make extra efforts to stand.