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Rosh Hashanah that falls on Friday evening:
Maariv: When [the first night of] Rosh Hashanah falls on Friday evening one begins the Maariv prayer from Mizmor Ledavid [psalm 29], [omitting all the Psalms from Lechu Neranina until Mizmor Ledavid]. [One recites the entire dialect from Mizmor Ledavid and onwards, including Ana Bekoach; all the stanzas of Lecha Dodi; Mizmor Shir, Kegavna. In Lecha Dodi the normal dialect of Berina is recited. After Shemoneh Esrei one recites Vayechulu, and Meiyn Sheva. Following the recital of Meiyn Sheva one recites Ledavid Mizmor and then Kaddish with Tiskabel. This Kaddish is then followed by “Mizmor Ledavid Hashem Roiy” as is usually recited on Friday night. One then recites half Kaddish, Barchu and Aleinu.]
Mieiyn Sheva: When Rosh Hashanah falls on Shabbos, the Chazan recites Mieiyn Sheva after Shemoneh Esrei of Maariv as is always done on Friday night. Hamelech Hakadosh is to be recited in place of Hakeil Hakadosh. [Likewise when the congregation customarily recites the paragraph of Magen Avos they say the words Hamelech Hakadosh in place of Hakeil Hakadosh.] The Chazan however concludes only with the blessing of Shabbos [i.e. Mikadeish Hashabbos] even if Rosh Hashanah or Yom Kippur falls on Shabbos.]
Kiddush: When [the first night of] Rosh Hashanah falls on Friday evening, the following passages prior to Kiddush are read in an undertone: shalom aleichim, eishes chayil, mizmor ledavid Hashem ro’i, da hi se’udasa.
If in Meiyn Sheva the Chazan said Hakeil Hakadosh instead of Hamelech Hakadosh must he repeat the Meiyn Sheva?
Practically one is not to repeat the blessing. See Halacha 5B in Q&A for the full details of this subject!
When Rosh Hashnah falls on Shabbos may one ask physical requests of G-d?
Yes. One may make requests on physical and spiritual matters on Rosh Hashanah even if it coincides with Shabbos.
When saying a Mi Shebeirach for the sick on Shabbos Rosh Hashanah is one to say Shabbos Hi Milizok?
Other customs: Some are accustomed to begin Maariv from after Lecha Dodi, by Mizmor Shir. [M”E 625/41]
 The reason: Some write the reason is because there is a Mitzvah of Simcha on Yom Tov and we hence desire to speed the conclusion of Maariv. [Otzer Minhagei Yeshurun p. 64; See Admur 270/1 regarding Bameh Madlikin that it is omitted on Yom Tov in order to hasten Simchas Yom Tov] Alternatively the reason is because these Psalms contain the words Rina, and on Yom Tov we emphasize the words Simcha. [Ketzos Hashulchan 77 footnote 13] Alternatively the reason is because the first five Mizmorim until Mizmor Ledavid relate to the five weekdays until Erev Shabbos while the psalm of Mizmor Ledavid relates to Erev Shabbos. Hence we omit the first five Zemiros as it is not proper to relate them to Yom Tov. [Sichas Kodesh 2 p. 121] To note that the Mitzvah of Simcha does not apply on Rosh Hashanah, and hence we recite Berina in Lecha Dodi as a normal Shabbos, thus the only reason that explains why on Rosh Hashanah we omit these Psalms is the last reason mentioned. [Glosses of Rav Raskin on Siddur p. 250]
 Shaar Hakolel 17/6; Ketzos Hashulchan 77 footnote 13; Otzer Minhagei Chabad p. 63
Other customs: Some are accustomed to only recite the first and last stanza of Lecha Dodi. [M”E 625/41] Others recite the first two stanzas of Lecha Dodi corresponding to Zachar and Shamor. [Alef Hamagen 625/56] Others recite the entire Lecha Dodi with exception to the stanza of Hisnaari Meiafar Kumi which is omitted. [Peri Megadim] Some are accustomed to omit Kegavna being that it mentions that all the other days are filled with wrath which is untrue regarding Yom Tov. [Siddur Yaavetz; Likkutei Mahrich Pesach; Alef Lamateh 625/67; Divrei Torah 9/72; Piskeiy Teshuvos 487/3]
 Sefer Haminhagim p. 52 [English]; Rebbe in Machzor Chabad; Luach Kolel Chabad [Edited]; Implication of instructions in Siddur Yaavetz; M”E 582/2; See Otzer Minhagei Chabad p. 62; Glosses of Rav Raskin on the Siddur
The reason: The Rebbe [in a footnote on the Machzor ibid] explains that Rosh Hashanah and Yom Kippur are not given for days of rejoicing [see Admur 582/10] and hence we do not change from the normal wording of Berina.
Other Yomim Tovim: By other Yomim Tovim the custom is to interchange the word Berina for Besimcha. [Hagahos Hasiddur of Rebbe Rashab; Ketzos Hashulchan 77/2; Mishnes Chassidim “Leil Yom Tov” 1/2].
Difference between Rina and Simcha: The term Rina denotes a bittersweet joy, a joy that comes as a result of a previous distance. However Simcha does not have any bitterness mixed with it at all. [Magen Avos Vayishlach’ Ketzos Hashulchan 77 footnote 13] Accordingly it is understood why on Rosh Hashanah we do not recite Besimcha, as there is bitterness involved in the repentance required.
Other customs: Some are accustomed to recite Besimcha even on Rosh Hashanah. [Minhag Rav Ahron of Belz, brought in Piskeiy Teshuvos 582 footnote 20]
 Mateh Efraim 582/2; Otzer Minhagei Chabad p. 63
 The above order is written in Piskeiy Hasiddur footnote 40; Otzer Minhagei Chabad p. 69
 Admur 582/4; See Halacha 5A for the full details of this Halacha!
 Sefer Haminhagim p. 118 [English]
 Mateh Efraim 584/25 [regarding an ill person]; Az Nidbaru 14/21; Piskeiy Teshuvos 582/7
 Although asking for one’s physical needs is ordinarily forbidden on Shabbos [and Yom Tov] [See Admur 294/1; 299/15; 188/4; 584/5] nevertheless it is permitted to do so on Rosh Hashanah being that the days was established to be a day of judgment which requires prayer and supplication. [ibid] Vetzaruch Iyun from the ruling in 584/5 regarding Avinu Malkeinu, that it is omitted on Rosh Hashanah due to Bakashas Tzerachav! Some suggest that only a set prayer of request was negated on Shabbos Rosh Hashanah while a private request is not only allowed but is motivated to be done. [Piskeiy Teshuvos 582 footnote 21]
Where in Shemoneh Esrei are the requests to be made? The requests are to be made after the first Yehiyu Leratzon but prior to reciting the second Yehiyu Leratzon. [See Admur 122/1; M”B 122/3]
 Mateh Efraim 584/25
 As Rosh Hashanah is the time to ask for blessings being that it is judgment day.
 Likkutei Mahrich Yom Kippur, brought in Piskeiy Teshuvos 582 footnote 21