When during davening is it correct to respond to the hearing of Hashem’s name with Baruch Hu U’Varuch Shemo? The rule of not responding when one desires to be yotzai the bracha does not seem to be applied consistently. Can you give specific examples? Thank you

A Brachah one is hearing to fulfill his own obligation with, he shouldn’t respond “Baruch Hu Uvaruch Shemo” as he should hear the whole Bracha. Similarly since this response has no source from the Talmud, if you are in a place you are not allowed to interrupt with talking, for instance, if you are in the middle of a Brachah, you should not respond “Baruch Hu Uvaruch Shemo” (Sulchan Aruch OC 124 9 and commentary ibid)