·до́брый - kind, good
·споко́йный - calm
·Добрый день! - Good day!/Good afternoon!
·Спокойный ночи! - Goodnight! (lit. ‘Calm night!’)
Does день stand in for morning, day, afternoon and evening? With ночи used only for the latest night? Are there more words for other times of day that we've just not met yet, or unless it's the late night, call it a день?
Couldn't it also be "добрый вечер", or is that strictly speaking only for the late evening?
Вечер starts when it begins to get dark. While this might overlap with what people call afternoon, I don't think I'd use this as a translation.
Note that "afternoon" has its own word. I tried putting it into "good afternoon" but it said it was wrong, and the right answer was the word for "day." DL please fix.
Russian doesn’t really have a word for ‘afternoon’. The most common ways to express the idea are «втора́я полови́на дня» ‘second half of the day’ or «[вре́мя] по́сле обе́да» ‘[time] after lunch’, but they don’t really work in the greetings.
And whether you'll use длброе or добрый depends on the word that comes after, right? If so, is the only way to know which one to use by memorisation?
Yeah, it depends on the word it's describing. There are a few rules to help you remember - words ending in consonants are masculine, words ending in -а or -я are feminine, words ending in -о or -е are neuter. Words that end in -ь can be either masculine or feminine and those ones need to be memorised. The adjective endings are more or less the same - for this example, добрый is masculine, добрая is feminine, доброе is neuter. It's quite easy to get used to the patterns with practice and time :)