Uterque discipulus codicem suum legit.

Uterque means each of the two or, in smoother English, sometimes both.

A very important point: it is used for two, not for three or more. So uterque discipulus means each student of the two students.

That is why this word often feels half-way between English each and both:

• each because it treats them one by one
• both because the total number involved is two

Because uterque is distributive: it looks at the two students individually.

So Latin says, in effect, each student reads his own book, not the students read.

That is why you get:

• uterque discipulus = each student not
• utrique discipuli here

Even though two students are meant in real life, the grammar is singular because the sentence treats them one at a time.

For the same reason: the grammatical subject is uterque discipulus, which is treated as singular.

So Latin uses:

• singular subject: uterque discipulus
• singular verb: legit

This is very natural in Latin. English often does something similar with each:

• Each student reads

So the Latin grammar matches that pattern.

Both can be translated as both, but they are not used in exactly the same way.

• uterque discipulus legit = each student reads
  This stresses the two students individually.
• ambo discipuli legunt = both students read
  This treats them more as a pair together.

So:

• uterque usually takes singular grammar
• ambo takes plural grammar

That is a very common distinction in Latin.

Because codicem is the accusative singular, the form used for the direct object.

The dictionary form is:

• codex = nominative singular

But in this sentence, the book is the thing being read, so it must be in the accusative:

• codicem

This comes from the third declension:

• nominative: codex
• genitive: codicis
• accusative: codicem

So codicem means the book / codex as the object of legit.

Because suum must agree with codicem, the noun it describes.

Here codicem is:

• masculine
• singular
• accusative

So the possessive adjective must also be:

• masculine
• singular
• accusative

That gives suum.

The key idea is this: suus, sua, suum agrees with the thing possessed, not with the possessor.

So:

• discipulus is the possessor
• codicem is the thing possessed
• therefore the correct form is suum

Because suus is the reflexive possessive. It refers back to the subject of the clause.

So in this sentence:

• uterque discipulus ... codicem suum legit means
• each student reads his own book

If you used eius, that would normally mean his book referring to someone else, not back to the subject in the same direct way.

So:

• suum = his own
• eius = his, often someone else's, not reflexive

That is why suum is exactly the right choice here.

Normally, yes.

Because the subject is uterque discipulus and the possessive is suum, the natural meaning is distributive:

• student A reads his own book
• student B reads his own book

So the sentence strongly suggests one book each, not one book shared by both.

If Latin wanted to make a shared book clearer, it would usually phrase that differently.

• uterque discipulus is nominative singular
• codicem suum is accusative singular

Why?

Because:

• the subject of a finite verb goes in the nominative
• the direct object goes in the accusative

So:

• uterque discipulus = the one who reads
• codicem suum = the thing read

This is one of the most basic sentence patterns in Latin: subject + object + verb

Yes. Latin word order is much more flexible than English word order.

So these could still mean roughly the same thing:

• Uterque discipulus codicem suum legit
• Uterque discipulus suum codicem legit
• Codicem suum uterque discipulus legit

The endings show the grammar, so the meaning does not depend as heavily on position as it does in English.

That said, the word order can affect emphasis:

• putting codicem suum earlier can highlight the object
• putting legit at the end is very common in Latin

So the original order is natural, but not the only possibility.

Yes, in ordinary spelling it can.

Without vowel marks, legit can represent:

• present: he/she reads
• perfect: he/she read

In texts with macrons, the difference is clearer:

• legit = present
• lēgit = perfect

In a real sentence, context usually tells you which one is meant. If the meaning has already been given to you as present, then you should understand it here as reads.

Yes, that is its normal meaning.

Uterque refers to two and only two people or things. If more than two are involved, Latin would use other words instead, depending on the sense:

• omnes = all
• singuli = one each
• quisque = each

So in this sentence, uterque tells you immediately that the sentence is about two students.