Baruch Hu Uvaruch Shemo:
A. The Mitzvah:
Any time one hears [G-d’s name mentioned in] any blessing the custom is to recite Baruch Hu Uvaruch Shemo. This is a [mere] custom and was not instituted by the Sages. [The saying of Baruch Hu Uvaruch Shemo has sublime mystical meaning.]
Is one to answer Baruch Hu Uvaruch Shemo every time he hears Hashem’s name mentioned?
Some Poskim rule one is to answer Baruch Hu Uvaruch Shemo every time he hears Hashem’s name mentioned, even not in a blessing. Other Poskim rule that it is only answered when mentioned in a blessing, and so is the custom.
Is one to answer Baruch Hu Uvaruch Shemo in Birchas Kohanim?
Some Poskim rule one is to answer Baruch Hu Uvaruch Shemo upon hearing Hashem’s name in Birchas Kohanim. Other Poskim, however, rule one is not to answer Baruch Hu Uvaruch Shemo [and so is the custom of many].
Is one to answer Baruch Hu Uvaruch Shemo when hearing the blessing of a Mitzvah or Birchas Hanehnin that he is not being Yotzei?
Some write that the custom is not to be particular to answer Baruch Hu Uvaruch Shemo for Birchas Hamitzvos or Birchas Hanehenin being that at times people hear these blessings in order to fulfill their obligation, in which case it is forbidden to answer Baruch Hu Uvaruch Shemo. Accordingly, the main time for answering Baruch Hu Uvaruch Shemo is by Birchas Hashachar, Chazaras Hashatz, and by an Aliyah.
Is one to answer Baruch Hu Uvaruch Shemo to Birchas Hatorah of an Aliyah?
If the person is saying the blessing quickly and answering Baruch Hu Uvaruch Shemo will cause one to miss the conclusion of the blessing, what is he to do?
In such a case it is better that he be quiet and not answer Baruch Hu Uvaruch Shemo, than to answer it and miss the conclusion of the blessing and end up saying Amen right after saying Baruch Hu Uvaruch Shemo.
Pronouncing the words properly:
One must be careful to recite each one of the words Baruch Hu Uvaruch Shemo properly. One is not to swallow any of words or connect them, such as to say Baruchu Uvaru Shemo, [or Berich Shemo] and the like. One is to be careful to recite the words immediately after hearing Hashem’s name and not while Hashem’s name is still being said.
 Admur 124:2 and 8; Ketzos Hashulchan 22:5
 Admur 124:8; Michaber 124:5; Levush 124:2; Tur 124 in name of Rosh 4:19
Other opinions: Some Gedolei Yisrael avoided saying Baruch Hu Uvaruch Shemo when hearing Hashem’s name recited in a blessing. [Gr”a in Maaseh Rav 43; See Piskeiy Teshuvos 124 footnote 101] Others are accustomed not to answer it during Chazaras Hashatz in order not to make an interval in the blessing, and treat it similar to Shemoneh Esrei. [See Peas Sadcha 19; Shulchan Hatahor 124:7; Chazon Ish; Piskeiy Teshuvos ibid]
 The reason: This wording is in light of the verse “Ki Sheim Hashem Ekra Havu Godel Lelokeinu”, that when one mentions Hashem’s name one is to give praise to Hashem. Furthermore, even when one mentions a Tzaddik of flesh and blood one is required to bless him as the verse [Mishlei 10:7] states “Zecher Tzaddik Levracha” that when you remember a Tzaddik you must bless him [Admur ibid]
 Admur ibid; Rishon Letziyon on Brachos 63 “It is thus not an obligation from the letter of the law”
 See Kaf Hachaim 5:12; Maaseh Rokeiach; Makor Chaim of Chavos Yair 124
 Elya Raba 124:9; Shiyurei Kneses Hagedola 128 Hashmatos; Birkeiy Yosef 124:7; Yosef Ometz 296 in name of Chareidim 4:17; See Poskim in next Q&A; See Ashel Avraham of Butchach 124
 Makor Chaim of Chavos Yair 124:5; Tehila Ledavid 124:3 in name of Dvar Shmuel 295 and that so is the custom; Magen Giborim 124:11; Piskeiy Teshuvos 124:11
 Shiyurei Kneses Hagedola; Peri Chadash 124:5; 128:13; Kaf Hachaim [Falagi] 15:76]; Ben Ish Chaiy Tetzaveh 15; Kaf Hachaim 124:27 that so is custom; See Ashel Avraham of Butchach 124 that this is done only by Nesias Kapayim to show the love for the Mitzvah
 Shalmei Tzibur p. 145 that so is implied from Dvar Shmuel 295, brought in Kaf Hachaim ibid; See Poskim in previous Q&A!
 See Piskeiy Teshuvos 124:11
 Mishneh Halachos 13:16
 Piskeiy Teshuvos 124 footnote 106
 Yifei Laleiv in Yosher Leivav 5; Kaf Hachaim 124:28; Shaar Hatziyon 124:24
 Chesed Lealafim 124:4; Kaf Hachaim 124:29
 Makor Chaim 124; Piskeiy Teshuvos 124:11