Regarding when they use characters for characters with foreing/fantasy names

[Kadmos1]

The "Japanese writing system" Wikipedia entry calls it a "mixed logographic (kanji), syllabic (hiragana, katakana), and alphabetic (rōmaji)".

Question when they use Greek names: Do they use a different character set when using traditional Greek or Latinized spellings? For example, in Sora no Otoshimono, the Romanization of Ikaros' name is spelled from the Greek spelling rather than the Latin of "Icarus" but Astraea is from a Latinized version of Astraia.

Also, when Japanese uses characters for more fantasy names (Slayers' Zelgadiss or Gourry, for example), how do they write it?

[R.LocK]

...Katakana.

[teejmo]

Yeah, Japan rarely ever uses "romanji" (the Latin Alphabet), and that's almost strictly for us foreigners who don't have a clue how to read their system. Of course, they learn the Latin alphabet, but that's so they can learn how to read foreign languages. For fantasy and foreign names (and for some Japanese names, though it is rare), they use katakana - which is modified characters that are used for foreign words, like bread (パン, pan, which is bread in... Portuguese or Spanish, can't exactly remember which), and foreign titles (names, companies, countries - though they do have kanji for some countries).
As to the Greek/Latin thing like from Sora no Otoshimona, that's probably a slight mistranslation mostly due to the names having hardly any difference in pronunciation (the Japanese "alphabet" is a syllabary, so they write it out as how it sounds).

[Honoakari]

Rather than mistranslation, perhaps the author simply chose the variants of said names that sound better when transposed into kana. IMO "Asutorea" sounds a bit more natural and easier to pronounce than "Asutoraiaa", and the vowel "o" makes "Ikarosu" sound a bit more majestic than "Ikarusu".