Magistra rogat uter discipulus hanc sententiam facilius intellegat.

Uter means which of the two. Latin uses it when the choice is specifically between two people or things.

So uter discipulus means which student (of the two).

If the teacher were asking among more than two students, Latin would more naturally use quis or another construction depending on context.

It is subjunctive because this is an indirect question.

The direct question would be something like:

• Uter discipulus hanc sententiam facilius intellegit?
  Which student understands this sentence more easily?

When that question is reported after a verb like rogat (asks), Latin normally puts the verb of the indirect question into the subjunctive:

• Magistra rogat uter discipulus hanc sententiam facilius intellegat.

So intellegat is not there because the action is unreal or doubtful; it is there because Latin marks indirect questions with the subjunctive.

Because rogat can introduce an indirect question directly.

Latin often uses verbs like rogare, quaerere, or sciscitari to report questions. After such a verb, the question follows in indirect form:

• rogat uter discipulus... intellegat
  she asks which student... understands

So rogat does not need any extra word meaning whether here, because the indirect question is introduced by uter (which of the two).

Discipulus is nominative singular.

It is the subject of intellegat in the indirect question:

• uter discipulus ... intellegat
  which student ... understands

Even though the whole clause depends on rogat, the student is still the one doing the understanding, so discipulus stays nominative.

Because it is the direct object of intellegat.

The verb intellegere takes a direct object: you understand something.

So:

• hanc = accusative feminine singular of hic, haec, hoc
• sententiam = accusative feminine singular of sententia

They agree with each other and together mean this sentence.

Facilius is the comparative adverb of facile (easily).

• facile = easily
• facilius = more easily

So it modifies intellegat and tells us how the student understands the sentence.

It is adverbial, not adjectival. In other words, it means:

• understands more easily

not

• understands an easier sentence

The comparative adverb facilius means more easily, but Latin does not always state explicitly what it is being compared with.

In this sentence, the idea is something like:

• which of the two students understands this sentence more easily

The comparison is most naturally between the two students: one understands it more easily than the other.

Latin often leaves that comparison implicit because the context already makes it clear.

Because this is not an indirect statement; it is an indirect question.

Latin usually uses:

• accusative + infinitive for reported statements
• subjunctive for indirect questions

Compare:

• Magistra dicit discipulum hanc sententiam intellegere.
  The teacher says that the student understands this sentence.

but

• Magistra rogat uter discipulus hanc sententiam intellegat.
  The teacher asks which student understands this sentence.

So the infinitive would be the wrong construction here.

Latin word order is more flexible because the endings show the grammatical relationships.

English depends heavily on word order:

• The teacher asks which student understands this sentence more easily.

Latin can move words around more freely:

• Magistra rogat uter discipulus hanc sententiam facilius intellegat.

This order is perfectly normal. A few things to notice:

• Magistra comes first as the main subject.
• rogat introduces the indirect question.
• uter discipulus appears early because it is the key question word.
• intellegat comes at the end, which is very common in Latin.

So the order may feel unusual to an English speaker, but it is idiomatic Latin.

No, not here.

Uter asks which one of two. It identifies one member of a pair, not a type or category.

So uter discipulus means:

• which student (of the two)

not:

• what sort of student

If Latin wanted to ask about a type or quality, it would use a different word.

In this context, yes.

The Latin noun sententia can have several meanings depending on context, such as:

• opinion
• thought
• judgment
• sentence

Here, because the object is something being understood in a classroom context, sentence is a very natural meaning.

So hanc sententiam is best taken as this sentence.

Intellegat is present subjunctive.

It is present because the main verb rogat is also present, and the indirect question refers to an action that is viewed as happening at the same time:

• The teacher asks which student understands...

So the tense is not chosen just by time in an English sense; it also follows Latin’s normal sequence in indirect questions.

Yes. That is exactly the kind of direct question behind the indirect one.

Direct question:

• Uter discipulus hanc sententiam facilius intellegit?

Indirect question:

• Magistra rogat uter discipulus hanc sententiam facilius intellegat.

The main change is that intellegit becomes intellegat because the clause is now indirect.

By itself, rogat simply means asks. The sentence does not say whom she asks.

Latin could add a person if needed, for example:

• Magistra puerum rogat...
  The teacher asks the boy...

But in your sentence, the focus is not on the person being asked. The important part is the content of the question: which student understands the sentence more easily.