I noticed that “Baruch Shem K’vode Malkhuto L’Olam Voh’ed” is said at the end of the: 1)Rasheet Khakhmah… morning arising prayer when a portion of our souls are returned 1/64th? 2) after the Shema Yisrael… 3) at the end of Anah K’khoachk said as part of Friday night Kabbalat Shabbat prayers 4) I believe when we perhaps accidently say a prayer in vane and if I recall correctly 5) After the b’racha made over the head Tefillin in case the b’racha over the arm Tefillin actually covered the head Tefillin as well and 6) out loud at the end of Yom Kippur. What are the reasons and the laws/guidelines why we say this for each of 1-6 above and who are the key rabbonim who made these decisions’ra and is no. 4 (see above) appropriate? Are there other reason/s and places it is said and is there a rule of thumb for it to be appropriately said?
You asked a very comprehensive question in trying to organize the basis behind "Baruch Shem" in it's different places. I will begin with saying "Baruch Shem" after Shema. The Gemara (Pesachim 56a) tells us that it was Yaakov our forefather who was the first to utter this phrase. Yaakov was in doubt before his death of the faithfulness of his children to the belief in one G-D. When his children said "shema Yisrael, etc. and proclaimed their faithfulness in G-D, Yaakov answered in response "Baruch Shem etc.". Meaning, that Yaakov in return of hearing their acceptance of the Heavenly kingdom also blessed the kingdom of G-d. To commemorate this reciprocal acceptance of the Heavenly kingdom we continue to do so to this very day. However, the Talmud says although Yaakov said it, it was mentioned in the torah by Moshe therefore with all its importance, "Baruch Shem etc." should be said softly. The mentioning of "Baruch Shem " at the end of "Ana B'choach" is said for Kabbalistic reasons in order to reach a certain amount of letters. It is brought in Kabbalat Shabbat and also the Korbanot before Shacharit. If we say Hashem's name in vain, for example we mistakenly said the wrong Bracha the custom is to say "Baruch Shem ". Perhaps the reason being that since Hashem's name was already said we should accept upon ourselves the Heavenly kingdom. (Aruch Hashulchan 206). Saying The Baruch Shem out loud on Yom Kippur has a different reason. There is another Rabbinic source (Devarim Rabba 2) which explains the source of the Baruch Shem that Moshe Rabenu had heard the angels saying it while he was in Heaven and Moshe divulged this prayer to us. Therefore on yom kippur when aspire to emulate the angels (not eating, standing, wearing white etc.) we are entitled to say it out loud. (Tur 619) I hope I have answered your questions. Shabbat Shalom