Avia puellam ad cubiculum ducit, ut melius dormiat.

Because they have different jobs in the sentence.

• avia is the subject: the person doing the action.
• puellam is the direct object: the person being led.

In Latin:

• avia is nominative singular.
• puellam is accusative singular.

So the endings help show who is doing the action and who is receiving it, even if the word order changes.

Because ducit takes a direct object, and Latin normally puts the direct object in the accusative case.

The base word is puella = girl.
Its accusative singular form is puellam.

So:

• puella = the girl as subject
• puellam = the girl as object

In this sentence, the grandmother is leading the girl, so puellam is required.

Because it follows the preposition ad, and ad takes the accusative case when it means to or toward a place.

So:

• ad cubiculum = to the bedroom

This is a very common pattern in Latin:

• ad urbem = to the city
• ad scholam = to school
• ad cubiculum = to the bedroom

Here the accusative does not mean direct object; it is accusative because the preposition ad requires it.

Ducit is from ducere, which often means to lead, to take, or to guide.

In this sentence, it means something like:

• leads
• takes
• guides

So avia puellam ad cubiculum ducit is literally something like the grandmother leads/takes the girl to the bedroom.

Grammatically, ducit is:

• 3rd person singular
• present tense
• active voice
• indicative mood

So it means she leads/is leading.

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

English depends heavily on order:

• The grandmother leads the girl ≠ The girl leads the grandmother

Latin can move words around more freely because:

• avia shows the subject
• puellam shows the object

So this sentence could be rearranged in several ways without changing the basic meaning, for example:

• Avia puellam ad cubiculum ducit
• Puellam avia ad cubiculum ducit
• Ad cubiculum avia puellam ducit

The normal order here is perfectly natural, but it is not the only possible one.

Here ut introduces a purpose clause.

That means it explains why the grandmother leads the girl to the bedroom:

• ut melius dormiat = so that she may sleep better / in order that she sleep better

This is one of the most important uses of ut in Latin.

A very common pattern is:

• main verb + ut + subjunctive

So in this sentence:

• ducit = main verb
• ut ... dormiat = purpose clause

Because after ut in a purpose clause, Latin uses the subjunctive, not the indicative.

So:

• dormit = she sleeps / is sleeping (indicative, plain statement)
• dormiat = she may sleep / might sleep (subjunctive)

In other words, the sentence is not just stating that she sleeps better; it gives the purpose of taking her to the bedroom.

This is a key rule:

• ut + subjunctive = often so that ...

The subject of dormiat is understood from the context: it is puella, the girl.

Latin often leaves pronouns unstated when they are clear.

So the idea is:

• The grandmother leads the girl to the bedroom
• so that the girl may sleep better

It would not naturally mean that the grandmother is the one meant to sleep better.

Because melius is the comparative form of bene.

• bene = well
• melius = better

Since the sentence means that she will sleep better, Latin uses melius.

This is like:

• bonus = good
• melior = better
  and similarly
• bene = well
• melius = better

So melius dormiat means may sleep better.

It is a purpose clause.

Why? Because it expresses the goal or intention behind the action:

• The grandmother takes the girl to the bedroom in order that she sleep better.

A result clause usually describes what actually results and often has clues such as:

• tam, ita, sic, adeo, and so on in the main clause

There is no such signal here, and the meaning clearly expresses purpose, so this is best understood as a purpose clause.

Not exactly in the way English tense does.

In a purpose clause, the present subjunctive usually shows action that is contemporaneous with or future relative to the main verb.

Here the main verb is ducit (she leads/is leading), and the sleeping is the intended result of that action. So dormiat means something like:

• so that she may sleep
• so that she can sleep

It is not mainly about absolute present time; it is about the purpose connected to the main action.

Not in this sentence, if the meaning is to the bedroom.

Latin usually needs ad + accusative for motion toward something in this kind of expression:

• ad cubiculum = to the bedroom

Without ad, cubiculum by itself would not naturally express that same idea here.

Some place expressions in Latin can show motion without a preposition, but that mostly happens with certain special kinds of place words, such as names of cities, small islands, and a few common nouns like domus. Cubiculum is not one of those special cases here.

Yes. Latin often leaves out words that English would normally say explicitly.

Two important examples here are:

• the subject pronoun she with ducit
• the subject pronoun she with dormiat

Latin does not need to say ea ducit or ea dormiat, because the verb endings already show third person singular.

So learners should get used to the idea that Latin often expresses with endings what English expresses with separate words.