When is it expected to write someone's name correctly?
I think "correctly" is the wrong word to use here, and it will cloud the issue. What's "correct" for a name is somewhat ill-defined (I've known a Risa and a Lisa who both had the same Japanese name. Which of them was incorrect?). The real standard is to spell in accordance with the person's wishes.

As a German person, the blanket assertion "the German name "Müller" is expected to be spelled correctly" is not in my experience true. Rather I would say, as the ease of writing the umlaut electronically increases, the likelihood that someone will be unhappy that you didn't bother to make the minimal effort increases too. There are still plenty of systems that only take ascii characters and people will make do with Mueller or even (sigh) Muller. Japanese people know quite well that you probably couldn't read "田中" nor "タナカ", and they will themselves sign their correspondence to you "Tanaka". They won't expect you to make the effort to learn how to read and type kanji, because they know it's exceptionally more difficult than finding a diacritic. (They will however be quite happy if you do manage.)

For citations, the issue is a little different: it is expected that you cite the authors as written on the paper. If you are citing an English-language paper, it is very likely that you will be citing "Ryouichi Tanaka and Robert Müller", because that's what they called themselves. If you are citing a Japanese manuscript, you are expected to cite "田中良一　ロバート・ミュラー". Maybe as "田中良一　ロバート・ミュラー (Ryouichi Tanaka and Robert Müller)" if you are yourself writing in English.

To sum up: 


When is it expected to spell someone's name correctly? Always, but correctly ≠ in the original alphabet, or following some specific romanization rules.
When is it expected to spell someone's name in the original alphabet? When you can reasonably do so without major effort, and it does not hinder communication.