[quote]Though I'm interested in which anime had that last trope of the girl being inhuman from her singing on a high-elevated platform.[/quote]

[hider=SPOILERS][url=http://youtu.be/u9pDptOdahk?t=4m30s]Kaze no Stigma[/url] - She's a clone with a limited life span produced solely for the purpose of sacrifice.

[url=https://www.youtube.com/watch?v=qmlTpjonS00]Death Note[/url] - She has Shinigami eyes. At the ending, Misa is singing on a balcony. There's no moonlight, but a sunset is close.

[url=https://www.youtube.com/watch?v=2B-mHS1a9tM]Deadman Wonderland[/url] - She's the Wretched Egg. No moonlight in this one. Perhaps elevated female singing is sufficient for the trope to hold.

[url=https://www.youtube.com/watch?v=LF11Wf5te2I]Guilty Crown[/url] Weird magicky stuff about her that's hard to explain because Guilty Crown's plot barely makes sense. But trust me, not 100% human. [/hider]

So, it happens quite a lot.