When did Opening Credits become Closing Credits?

Well, a quick Wikipedia check produced this nugget, which answers part of the question:

The use of closing credits in film to list complete production crew and cast was not firmly established in American film until the 1970s. Before this decade, most movies were released with no closing credits at all. Films generally had opening credits only, which consisted of just major cast and crew, although sometimes the names of the cast and the characters they played would be shown at the end, as in The Wizard of Oz, Mary Poppins, Oliver! and the 1964 Fail Safe. Two of the first major films to contain extensive closing credits – but almost no opening credits – were the blockbusters Around the World in 80 Days (1956) and West Side Story (1961). West Side Story showed only the title at the beginning of the film, and Around the World in 80 Days, like many films today, had no opening credits at all.

And this came from a separate Wikipedia page, which answers the "Extra Credit" portion of the question:

The ascendancy of television movies after 1964 and the increasingly short "shelf-life" of films in theaters has largely contributed to the credits convention which came with television programs from the beginning, of holding the vast majority of cast and crew information for display at the end of the show.

From what I have seen in other movies.stackexchange.com questions, the transition from the "studio system" (actors, etc., directly under contract to the studios) to a more independent system (late 50s/early 60s) probably contributed to this a lot. See Why do movies still have credits?