This question brought to you in part by the edition of "Hark the Herald Angels Sing" I was practicing this morning.
It lists the work as by Mendelssohn-Bartholdy, but I only knew the composer as Mendelssohn growing up (and I was with the Western Conservatory of Music, so they made you recite the composer's name, birth and death date, etc., for exams) and only seemed to encounter the hyphenated form later--it still seems the composer is most commonly called Mendelssohn.
I know he was born Jewish and "adopted" into a Catholic family for what I believe were reasons of education and opportunity, but I've never been clear on whether Mendelssohn was his birth name, or the name of the family that took him in. This then, is the root of the question: Which name was he generally known by, why, and what implications did that have, given longstanding bigotry, discrimination and outright hatred toward the Jewish people and anyone known to have Jewish decent? Does using one name or the other have, for lack of a better term, political connotations? I'd sort of taken the hyphenation as representing a reclamation of the composer's heritage, but if Mendelssohn was his birth name, I'm clearly wrong. So what implications are there of this change and when did it occur?
I'll also just take the liberty of paging /u/erus and /u/caffarelli, since I'm likely well within your bailiwick on this one.
To my knowledge, Mendelssohn wasn't adopted by another family, and actually his family was reasonably affluent and had a reputation--Felix and Fanny's grandfather was the philosopher Moses Mendelssohn. But yes, for political reasons, and social ones really, they changed the name to Bartholdy, but Felix retained the Mendelssohn. This is from the Grove Dictionary of Music:
In 1812 the Prussian chancellor, Hardenberg, issued a decree (the Gleichstellungsgesetz, or emancipation act) extending further rights to Prussian Jews. Nevertheless, on 21 March 1816 the Mendelssohn children were secretly baptized into the Protestant faith and Felix was given the additional names Jakob Ludwig. But not until 1822 did Lea and Abraham Mendelssohn become Protestants. At the same time they added the surname Bartholdy (after the precedent of Lea's brother Jakob Salomon, who had converted to Christianity several years before and adopted Bartholdy, the name of a family dairy farm, as a surname; Jakob advised Abraham to use the name Mendelssohn Bartholdy ‘as a distinction from the other Mendelssohns'). In explaining his decision to his daughter Fanny, Abraham Mendelssohn noted in 1820 that the Christian faith ‘contains nothing that can lead you away from what is good, and much that guides you to love, obedience, tolerance, and resignation, even if it offered nothing but the example of its Founder, understood by so few, and followed by still fewer’.
That answers where the name came from, and reading between the lines here, you can see the political implications. As you can glean, discrimination was a consideration in embarking on a career as a public figure in the 19th century, so covering as a Protestant probably would help. I'm not a Mendelssohn scholar, so I can't go much deeper than that with him here. But I will add that, aside from Wagner's famous anti-semitism (which was not as shocking in the 19th century as it is to us now), Schumann had some pretty anti-semitic things to say, frequently recording in his journals thoughts that aren't exactly kind, this despite his being quite good friends with several prominent Jews. Schumann was also a huge fan of Mendelssohn's; he loved his music and Mendelssohn lived up to many of the ideals that Schumann held for the future of music. Among these things, Schumann admired Mendelssohn's activity in "early" music (today we think of this as anything before the Baroque, but in the 19th c. it was anything more than 50 years old), in particular Mendelssohn's famous rediscover of Bach and promotion of him with a performance of the St. Matthew Passion. In any case, Schumann, who was a loud voice in music in the mid-19th century, was a big supporter of Mendelssohn, despite his Jewish heritage. We can speculate about what Schumann's opinion of Mendelssohn would have been if he had no Jewish heritage, but that's sort of felicitous.
I'm not answering all of your question here, since it's pretty broad and encompasses things I'm not an expert on, but hopefully that's at least somewhat helpful!