En julio mia kuzino volas lui ĉambron apud la maro por unu semajno.

Because en julio is a time expression meaning in July. Esperanto often puts time expressions at the beginning of a sentence if they set the scene.

So this order:

En julio mia kuzino volas lui ĉambron...

is very natural and means something like:

In July, my cousin wants to rent a room...

You could also move it later:

Mia kuzino volas lui ĉambron apud la maro en julio.

That would still be correct.

Month names normally do not take la in Esperanto.

So:

• en julio = in July
• en aŭgusto = in August
• en decembro = in December

Using la here would usually sound unnatural.

In Esperanto, possessive words like mia, via, ŝia, nia usually make the noun definite by themselves, so la is normally omitted.

So:

• mia kuzino = my cousin
• lia domo = his house
• nia instruisto = our teacher

Using la mia kuzino would usually be wrong in a normal sentence.

Yes. Kuzino means female cousin.

The ending -ino marks a female person:

• patro = father
• patrino = mother
• frato = brother
• fratino = sister
• kuzo = cousin / male cousin
• kuzino = female cousin

So the sentence clearly tells you the cousin is female.

Because volas means wants, and after voli Esperanto uses an infinitive to show what someone wants to do.

So:

• volas lui = wants to rent
• volas manĝi = wants to eat
• volas iri = wants to go

Here:

• volas is the main finite verb
• lui is the infinitive, meaning to rent

This works very much like English wants to rent.

Lui means to rent or to hire.

In this sentence, lui ĉambron means to rent a room.

Sometimes learners wonder whether lui means renting something from someone or renting something to someone. By itself, it can depend on context, but if needed Esperanto can make it clearer:

• lui de iu = rent from someone
• lui al iu = rent to someone

Here the context clearly means rent a room for herself.

The -n marks the direct object.

The thing being rented is the room, so ĉambro becomes ĉambron.

Compare:

• Mi vidas la domon. = I see the house.
• Ŝi aĉetas libron. = She buys a book.
• Mia kuzino volas lui ĉambron. = My cousin wants to rent a room.

So -n here shows what the action of lui is acting on.

Apud is a preposition meaning beside, next to, or by.

So:

• apud la maro = by the sea / next to the sea

The noun maro does not take -n here because it is the object of the preposition apud, not the direct object of the verb.

Also, la maro is natural because it means the sea, a specific kind of location, just as English says by the sea.

Because the phrase refers to the sea as a known location, not just a sea in a vague sense.

In Esperanto, as in English, that often makes the definite article natural:

• apud la maro = by the sea
• ĉe la rivero = at the river
• en la urbo = in the city

Without la, apud maro would sound less normal in this sentence.

Yes, dum unu semajno would also be possible, but there is a small difference in feeling.

• por unu semajno = for one week, with an idea of intended duration or booking period
• dum unu semajno = for one week, during one week, more neutral about duration

In a sentence about renting a room, por unu semajno is especially natural because it sounds like the planned rental period.

So this sentence suggests:

She wants to rent a room by the sea for a one-week period.

Because por is already a preposition, so semajno does not need the accusative -n.

After a normal preposition, the noun usually stays without -n:

• por unu semajno
• en julio
• apud la maro

If you removed the preposition and used the accusative of duration, you could say:

• Mia kuzino restos unu semajnon. = My cousin will stay one week.

But in this sentence, por unu semajno is the correct structure.

Because it means one week.

With unu, the noun is singular:

• unu tago = one day
• unu semajno = one week
• unu ĉambro = one room

But with numbers greater than one, Esperanto uses the plural:

• du tagoj = two days
• tri semajnoj = three weeks
• kvar ĉambroj = four rooms

So unu semajno is exactly what you would expect.

Not completely. Esperanto word order is fairly flexible, because endings like -n help show the roles of words.

The given sentence is very natural:

En julio mia kuzino volas lui ĉambron apud la maro por unu semajno.

But other orders are also possible, for example:

• Mia kuzino volas lui ĉambron apud la maro por unu semajno en julio.
• Mia kuzino en julio volas lui ĉambron apud la maro por unu semajno.

The original order sounds smooth because it starts with the time setting and then gives the main information.