[Shut She’eilat Shlomo 1:235]
One should not say "Baruch Hu U-Varuch Shemo" (Blessed is He and Blessed is His Name) during the blessing over the Shofar, because this blessing is also recited in order to fulfill his (the listener’s) obligation, and "Baruch Hu U-Varuch Shemo" is considered an interruption (Shulchan Aruch, Orach Chaim 585). This is the general principle: any time during which it is forbidden to interrupt, one should not say "Baruch Hu U-Varuch Shemo." As, for example, during the blessing before the Shema in the morning and evening, during Baruch She-Amar and during Yishtabach. This also applies any time that a blessing is recited in order to fulfill one’s obligation, as in Kiddush and Havdalah. But one may say "Baruch Hu U-Varuch Shemo" during the morning blessings, during the repetition of the prayer on Shabbat Evening in Me’ein Sheva, during Magen Avot and during the blessings for an Aliyah to the Torah.
6. Bowing in Shul
When one bows on a stone floor, he should spread out a towel, handkerchief or piece of paper (but not a Talit bag, since it is disrespectful to put it on the floor). We do this because it is written in the Torah (Vayikra 26:1): "Nor shall you place a figured stone in your Land to bow down on it." This means that it is forbidden to bow down, even to Hashem, on a stone floor (except for one in the Temple. Rama in Shulchan Aruch, Orach Chaim 131:8). And because all of this refers to when one’s face touches the ground, one should spread out the handkerchief in the place where one’s face will be and not under the knees (Mishnah Berurah #40-41. Luach Heichal Shlomo).
Question: Can women fall on their knees during Musaf on Rosh Hashanah and Yom Kippur, since women were not in the Temple courtyard and therefore would not bow? [In the book "Nefesh Ha-Rav" (pp. 214-215), Rav Soloveitchik rules that women should not prostrate on account of this reason].
Answer: Yes, they may. This is a Minhag. As long as there is a modest place and she does not have a bad back, a woman may prostrate.
7. Preparing on the First Day for the Second Day
Question: Is it permissible to prepare on the first day of Rosh Hashanah for the second day?
Answer: Regarding this issue, each day is considered as a holy day on its own, and just as we do not prepare on Shabbat for a weekday, so too we do not prepare on the first day of Rosh Hashanah for the second day, since there is room to say that perhaps the holiness of the second day is in fact a weekday in relation to the holiness of the first day (Shemirat Shabbat Ke-Hilchata p. 20 #58 and p. 59 #1). It is therefore forbidden to cook, wash dishes, prepare candles in the candlesticks, make beds, set tables, etc. (ibid. 1, 58) on the first day in preparation for the second day. One must wait until after the stars have come out.
One may, however, prepare on Rosh Hashanah for Shabbat when they are consecutive days, by making an Eruv Tavshilin (by setting aside, before the holiday begins, an eruv consisting of a cooked portion of food and a piece of bread or matzah. This is considered the start of the food for Shabbat and any further preparation of food is seen as a continuation of it, and is thus permitted).
Addition to the revised edition
If Rosh Hashanah falls on a Thursday and Friday, it is forbidden to prepare on Thursday for Shabbat even with an Eruv Tavshilin. The preparation may only be performed on Friday (Shut Yehaveh Daat 6:32).
Do not forget the essence of Rosh Hashanah: to perform Teshuvah, both in commandments between a person and Hashem and in commandments between a person and his fellow man.