well, we don't know where they have gone, and i don't expect to see them any time soon...
about the sunset-sunrise, i have two theories:
A. they are just non-convetional terms to describe the status of the sun, like using an archaic pronounciation, so Inane chose to leave it that way.
B. since their orginaztion name has the word "tsuki" (moon, the full name is either evil moon or red moon) in it, it's possible that they use the statue of the sun as mission codenames, like how the sound used "kiminamaro"
and last thing, about names.
people call each other by both, and it's possible to add suffixes to either the first or the family name, but since the name Hoshigaki was mentioned about only one time, it'd be much better to use Kisama if you want people to know what your'e talking about