Why is "Baruch hu ubaruch shemo" omitted during the reciting of certain brachot such as meggilah and shofar?
The cases you've raised are cases where someone is saying the beracha with someone else in mind, i.e. saying it on their behalf so they don't say it themselves. In fact, it is as if they are actually saying it themself.
Therefore, if they were to say "baruch Hu uvaruch Shemo" it is as if they had said "Baruch Atah Hashem baruch Hu Uvaruch Shemo Elokenu Melech HaOlam Boreh Peri Hagefen", which is not the correct way to say the beracha (Responsa Devar Shmuel).
This constitues an interruption to the blessing and this is not allowed.
Many great Poskim rule in accordance with the Devar Shmuel. See this page for more information.
In the context of your secondary question, essentially, you are trying to understand the distinction between He and His name. That when the response is made, it emphasizes through the connecting letter Vav that G-d (אתה, הוא) and His name (יהוה) are one.
Considering the common link between Megillat Esther and blowing the Shofar on Rosh HaShanah which you bring as an example of when we do not answer blessed is He and His name in response to reciting a blessing which contains the phrase, ברוך אתה יהוה (meaning blessed are You (G-d transcendent of the aspect of letters and names, accompanied with G-d's name), it would seem that those specific blessings which omit our response emphasize the idea of G-d himself מהותו, at His essence, which transcends even the level of (the letters of) His name.
There is no usage of any of G-d's names (none are written) in Megillat Esther, even though it is understood that the Holy One, blessed be He performed that miracle. The Megillah emphasizes G-d transcendent of all names.
In this sense, it also emphasizes the correlation between Purim פורים, (the day when Megillat Esther is read and Yom HaKippurim יום הכיפורים, (which is referred to as like Purim) which concludes with the fifth prayer service of Neilah, which comprises repeated recital of the 13 Attributes is Mercy and the culmination of Teshuvah Shleimah and the transformation of the five Gevurot, (Gevurah to Chesed) via the five Chassidim. The red thread tied around the Altar transforms into a white thread.
So too, in regard to Rosh HaShanah, the emphasis of the "sound of the Shofar" (קול שופר) is that the sound, without letters, is what introduces the names of G-d (עלה אלהים בתרועה יהוה בקול שופר).
In practice, the קול שופר, Tekiah (the simple sound, even without the divisions, meaning the Gevurot which comprise Shevarim and Teruah,) is both before and after the sounding of שברים-תרועה meaning the Chassidim, Chesed precedes, introduces and surrounds the aspect of Gevurah related to the existence of letters.
That (שברים-תרועה) is the concept of the five Gevurot (מנצפ״ך) which correspond to the five final letters, which according to Sefer Yetzirah 2:3 originate from the five parts of the mouth and give rise to the letters which make up all words and names. That מנצפ״ך is gematria: עולם חסד יבנה (Tehillim 89:3). Like Rambam explains in Moreh Nevuchim 3:53:2, this pertains to His Middot, which is the source of the concept the Sephirot.