Uter liber Graecus est, uter Latinus?

Uter means which of the two? It is used when the speaker is choosing between exactly two things.

So Uter liber Graecus est, uter Latinus? assumes there are two books, and the speaker is asking which one is the Greek book and which one is the Latin book.

A more general question word like quis or qui would not specifically suggest a choice between two.

It is repeated to match the two parts of the contrast:

• uter liber Graecus est = which book is Greek
• uter Latinus = which one is Latin

English often does the same thing: Which book is Greek, and which is Latin?
The repetition makes the pairing very clear.

Because they agree with liber, which is a masculine singular noun.

• liber = book, masculine singular
• Graecus = Greek, masculine singular
• Latinus = Latin, masculine singular

In other words, the adjectives match the noun they describe.

They are also in the nominative case because they are used with est:
liber Graecus est = the book is Greek

Yes, but it is only omitted because it is easy to understand from the first half.

The full version would be:

Uter liber Graecus est, uter Latinus est?

Latin very often leaves out a repeated verb when the meaning is obvious. This is called ellipsis. English can do something similar: Which book is Greek, which Latin?

Because Latin often avoids repeating words that are already clear from the context.

In the second half, liber is understood:

• uter liber Graecus est
• uter liber Latinus est

But instead of repeating liber, Latin simply says uter Latinus.

So the second uter is effectively standing for which book.

In the first half, uter is being used like an interrogative adjective because it directly modifies liber:

• uter liber = which book of the two

In the second half, uter functions more like a pronoun, because the noun liber is omitted but understood:

• uter Latinus = which one is Latin, meaning which book is Latin

So in this sentence it behaves in both ways.

Latin word order is much freer than English word order.

Interrogative words such as uter often come first, so Uter at the beginning is very natural. The rest of the sentence can then follow in a fairly flexible order.

The given order is straightforward and idiomatic:

• Uter = which of the two
• liber = book
• Graecus = Greek
• est = is

English depends heavily on word order for grammar, but Latin depends much more on endings.

Grammatically, Graecus could have different shades of meaning depending on context. It can mean something like:

• Greek by origin
• Greek in style or character
• written in the Greek language

In this sentence, because Graecus is contrasted with Latinus, the most natural meaning is written in Greek versus written in Latin.

So the contrast is probably about language, not nationality.

Not for exactly the same kind of question.

Utrum ... an ... is used for an either-or question, like:

Utrum hic liber Graecus est an Latinus?
= Is this book Greek or Latin?

But uter asks which one of the two?

So if you are identifying one book out of two, uter is the right word. If you are asking whether something is one thing or another, utrum ... an ... is the usual pattern.

Yes. That would be grammatically correct and completely clear.

But Latin often prefers to leave out repeated words when they are obvious. So the shorter version is more economical:

Uter liber Graecus est, uter Latinus?

That shorter form is natural Latin style, not bad grammar.