Answer: Regarding the obligation to rise while answering “Baruch Hashem Ha’Mevorach,” there are varying customs among the Sephardic and Ashkenazi communities, for Ashkenazim customarily stand during Barechu while the Sephardim customarily remain seated. There are some Ashkenazi communities who customarily rise even when responding to the Barechu recited by one receiving an Aliyah to the Torah and not only when it is recited by the Chazzan. On the other hand, Sephardim are not concerned with this at all and the greatest Sephardic luminaries have never observed this custom.
Regarding bowing while reciting “Baruch Hashem Ha’Mevorach,” this custom is indeed mentioned by several Rishonim including the Machzor Vitri, Orchot Chaim, and others. Hagaon Mishnah Berura quotes the Magen Giborim who has difficulty finding a viable source for this custom. The Mishnah Berura himself (in Be’ur Halacha, Chapter 113) brings a reference to this custom from a verse in Chronicles I (Chapter 29) which states: “David said to the entire congregation: ‘Bless Hashem, your G-d’; the entire congregation then blessed Hashem, G-d of their fathers, and they prostrated themselves and bowed to Hashem and to the king.” Various Acharonim quote other sources for this custom. Nevertheless, Hagaon Harav Yair Bachrach zt”l, author of Mekor Chaim, writes that any addition to the order of bowing listed in the Talmud constitutes a prohibition of adding on to the enactments of our Sages. Similarly, the Shulchan Ha’Tahor writes likewise that one should not observe these extraneous bowings; the notation there states that there is no correct source for bowing during Barechu and it was instituted in error. Hagaon Harav Ben Zion Abba Shaul zt”l writes likewise that one should not bow during Barechu because of the prohibition of adding onto the enactments of our Sages.
Hagaon Harav David Yosef Shlit”a disproves this claim of adding onto the bowings enacted by our Sages, for we find several places where we customarily bow, such as while reciting “Va’Ya’avor Hashem Al Panav” based on the opinion of the Mekubalim, although our Sages never instituted such bowings; we do not find that the Poskim were concerned about adding onto the bowings enacted by our Sages in such instances.
Nevertheless, Sephardic and Middle Eastern Jews have never observed the custom of bowing during Barechu and this custom should be continued and safe-guarded. Maran Rabbeinu Ovadia Yosef zt”l likewise did not bow at all during Barechu. One should therefore not bow while reciting Barechu.