Learn Hardcore Latin: Magister rogat: "Cur discipuli hodie in schola non manent?" - The teacher asks: "Why do the students not stay in the school today?"

Magister – nominative singular masculine of magister, magistri (teacher); subject of rogat.
rogat – 3rd person singular present active indicative of rogo, rogare, rogavi, rogatum (he asks).
Cur – interrogative adverb (why), introducing a direct question.
discipuli – nominative plural masculine of discipulus, discipuli (students, pupils); subject of manent.
hodie – adverb (today), historically from hoc die = on this day; indeclinable.
in – preposition taking the ablative here, meaning in / at (location).
schola – ablative singular feminine of schola, scholae (school, class); object of in.
non – adverb (not), negating the verb manent.
manent – 3rd person plural present active indicative of maneo, manere, mansi, mansum (they stay / remain).

Why is magister in the nominative case and not something like magistrum?

Magister is the subject of the verb rogat.
Latin uses the nominative case for the subject of a finite verb, so magister rogat means the teacher asks.
Magistrum would be accusative singular and would normally be used as a direct object, e.g. pueri magistrum vident – the boys see the teacher.

Why is discipuli nominative plural and not accusative plural (discipulos)?

Inside the quoted question Cur discipuli hodie in schola non manent?, discipuli is the subject of the verb manent.
The quoted part is a normal independent question, so it follows the usual pattern: nominative subject (discipuli) + verb (manent).
If this were an indirect statement like Magister dicit discipulos in schola non manere, then discipulos (accusative) would be the subject of the infinitive manere. But here we have a direct question, not an accusative-and-infinitive construction.

What exactly is rogat doing in this sentence, and why is there a colon and then a question?

Rogat means (he) asks and introduces what the teacher says.
The colon simply marks the beginning of the teacher’s direct speech, just as in English: The teacher asks: Why…?
So structurally, you have:

main clause: Magister rogat – The teacher asks
direct question: Cur discipuli hodie in schola non manent? – Why are the students not staying in school today?

In Latin, rogo can introduce either a direct question (as here) or an indirect question (with the verb in the subjunctive).

What does cur mean, and how is it used?

Cur is an interrogative adverb meaning why.
It normally stands near the beginning of a question and introduces both:

direct questions: Cur venis? – Why are you coming?
indirect questions: Rogo cur venias. – I ask why you are coming.

Unlike English, Latin does not need an auxiliary verb like do; cur plus a normal finite verb is enough.

What tense/aspect is manent, and how should it be translated into English?

Manent is present indicative, 3rd person plural: they stay / they remain.
Latin’s present tense covers both English simple present and present continuous, so manent can be translated as:

they stay / they remain
they are staying / they are remaining

Context decides which English form sounds more natural. In this sentence, are not staying is a good idiomatic translation.

Why use manent here and not sunt (are)? What is the difference?

Manent (from maneo) specifically means stay, remain, not go away / not leave.
Sunt (from sum) simply means are and does not carry the idea of staying.

Compare:

Discipuli hodie in schola non manent. – The students are not staying in school today (they leave or will leave).
Discipuli hodie in schola non sunt. – The students are not in school today (they are absent altogether).

So manent focuses on remaining there; sunt only states presence or absence.

What is the function and case of in schola, and how would it differ from in scholam?

In schola is in + ablative and expresses location: in school / at school.
With in, Latin uses:

in + ablative for place where (location): in schola manent – they stay in school
in + accusative for motion towards (into): in scholam eunt – they go into the school

So in schola = in (the) school, whereas in scholam = into (the) school.

Why is hodie in the middle of the sentence? Could it go somewhere else?

Latin word order is fairly flexible, especially for adverbs like hodie.
You could see:

Cur discipuli hodie in schola non manent? (as given)
Cur hodie discipuli in schola non manent?
Cur discipuli in schola hodie non manent?

All are grammatically correct. Moving hodie can slightly affect emphasis (for example, putting it early may stress today more), but the basic meaning remains the same.

Why does non come before manent? Can non go somewhere else?

In Latin, non usually stands immediately before the word it negates, and most often that is the main verb.
So non manent = they do not stay / they are not staying.

Non can also negate other elements if it is placed before them:

Non discipuli hodie in schola manent, sed magistri. – It’s not the students who are staying in school today, but the teachers.

In your sentence, the straightforward negation of the verb is best, so non manent is the normal position.

Could this be turned into an indirect question instead of a direct one? How would the Latin change?

Yes. An indirect question would drop the colon and quotation, and put the verb of the question into the subjunctive:

Magister rogat cur discipuli hodie in schola non maneant.

Here:

cur discipuli hodie in schola non maneant is a subordinate clause (indirect question),
maneant is present subjunctive (same time as rogat).

The English translation becomes: The teacher asks why the students are not staying in school today, without the quoted direct speech.

Can Magister rogat be translated as The teacher asked instead of asks?

Literally, rogat is present tense: asks / is asking.
However, Latin sometimes uses the present as a historical present when narrating past events. In such a narrative context, English may translate it with a past tense: The teacher asked…
If this sentence is just given on its own, outside a past-tense story, the most natural translation is The teacher asks: Why…? or The teacher is asking: Why…?