"It is nine thirty sharp."
If you think about it, the word "just" does have a similar meaning to "sharp" in English to talk about the precise time when something happened, as in "He's just arrived". I think "choudo" conveys a similar meaning in Japanese. That's why Duolingo shows "just" as a possible translation, even though the phrase does not mean "it's just X o'clock".
Pretty much, yup. When placed before a specific time or size or other measurement, ちょうど is more like "precisely", or "exactly".
The reason why "just" is listed as a possible translation is because ちょうど is also used in other contexts, not just the telling of time. You can say "私はちょうど [insert action here]", and it would be similar to "I just did ___"
I think you would say ちょうど食べたばかりです (choudo tabeta bakari desu) for "I just ate".
Used with ta-form of verb, followed by tokoro (or bakari) + da/desu
Choudo shigoto o oeta tokoro desu.
I’ve just finished the work.
Choudo ima tsuita bakari desu.
I've just arrived.
半 means half and 分 means minute, so 半分 could be seen as half a minute. When used alone after 時 it is understood as referring to the hour, much like in English. If you say "hours and a half" in English, most people will understand that as "and a half hour". Same thing here.
When you use です (desu) you need a noun or adjective attached to it directly with no particle. In this case it’s 9時ちょうど (9ji choudo) that goes with です so you can’t use a particle. You would use に to say what time an action is happening rather than a statement about what time it is.
I agree, using 15分前 is the closest equivalent to "quarter to". You can use it in other increments as well. 3時10分前 (3ji 10ppun mae) means "10 to 3 (2:50)".
I think if you say 3時15分すぎ, though, you actually mean "slightly after 3:15", so the nuance is a little different. The most natural way to say "quarter past 3" seems to be 3時15分 (3ji 15fun).