OG, I'll give my tentative answers to a couple of these questions -
I'm not sure why some Indian actors have gone by only one name, but I would say that it's only "some." Plenty had both names. I wonder if it might have to do with spreading celebrity across a large country with very many different languages and ethnicities ... but that's just a guess.
As for musical numbers replacing sex scenes - I'm not so sure. Indian popular cinema has, for a very long time, used the musical format (six songs, three dances - "masala" [mixture]) as its conventional structure, so it's not as if there were sex scenes in 1920s cinema that musical numbers replaced. Maybe in 1990s productions, perhaps, there would be a musical number where a Hollywood film would have shown a sex scene - but this would have more to do with cultural mores and censorship
as well as a long-standing musical tradition to use music, dance, etc., to suggest and evoke what can't be explicitly shown. As was often the case with classic Hollywood. I don't know anything about Indian filmmakers in general deciding to replace sex with music, but perhaps someone with better knowledge than myself can come up with examples.
Some recommendations for just starting out on popular Indian film -
Sholay (an incredibly entertaining 'curry western'),
Kaagaz Ke Phool,
Barsaat, in fact the films of Guru Dutt and Raj Kapoor in general ... as for a more recent one, the Tamil filmmaker Mani Ratnam is sometimes called something like the "Indian Spielberg." I haven't seen much but I very much love the film
Roja - you can see the wonderful opening song
here.