Labanakt

It is neutral. You can use it with your boss, your grandma, or your best friend.

No, that would just mean 'of the night' and sounds incomplete. Stick to 'Labanakt'.

'Labanakt' is the common, shortened version. 'Labos nakties' is the full, slightly more traditional form.

The most common response is simply 'Labanakt' or 'Labanakt ir tau' (Goodnight and to you).

Yes, it is the noun meaning 'a goodnight'. You don't say it to people, but you use it in sentences like 'Jis man palinkėjo labanakčio'.

Over time, the two words fused together in common speech and writing for efficiency.

Only if you are literally going to sleep at 6 PM. Otherwise, use 'Labas vakaras'.

It means 'Sweet dreams'. It's the most common follow-up to 'Labanakt'.

Not really, but 'Labanaktukas' or just 'Saldžių' (Sweet [dreams]) are used in very casual settings.

Only if the email is sent very late at night and you have a friendly relationship with the person.

It is a hard 'L' (velarized), similar to the 'L' in 'ball'.

It's grammatically correct but 'Labos nakties' is the fixed idiom. 'Geros nakties' sounds a bit like a literal translation from English.

It means 'Peaceful night'. It's a slightly more poetic or sincere way to say goodnight.

Yes, many Lithuanian lullabies and pop songs use it as a central theme.