A gennaio la figlia riceve un libro in regalo.

A gennaio means in January. Italian often puts a time expression at the beginning of the sentence to set the scene first.

So:

• A gennaio la figlia riceve un libro in regalo.
• literally: In January, the daughter receives a book as a gift.

You could also move it later:

• La figlia riceve un libro in regalo a gennaio.

Both are correct. Starting with A gennaio simply gives extra emphasis to the time.

With names of months, Italian usually uses a when meaning in a certain month:

• a gennaio = in January
• a marzo = in March
• a dicembre = in December

This is a very common pattern in Italian. English speakers often expect in, but with months Italian normally prefers a.

Italian uses the definite article much more often than English. So where English says the daughter only in certain contexts, Italian often naturally says la figlia.

Here, la figlia means the daughter. It suggests a specific daughter already understood from context.

Compare:

• la figlia = the daughter
• una figlia = a daughter

Figlia means daughter.

It is the feminine form related to:

• figlio = son
• figlia = daughter

So la figlia is the daughter.

Riceve is the third person singular present tense of ricevere, meaning to receive.

The subject is la figlia, which is she / the daughter, so the verb must match that subject:

• io ricevo = I receive
• tu ricevi = you receive
• lui/lei riceve = he/she receives

So la figlia riceve = the daughter receives.

Not always. It depends on context.

Ricevere can mean:

• to receive
• sometimes to get
• in some contexts to welcome or to admit

In this sentence, receives or gets both work in English, but receives is the more direct translation.

Un libro means a book, so it introduces the book as something non-specific or new information.

Compare:

• un libro = a book
• il libro = the book

If the sentence said il libro, it would sound like a particular book already known to the listener or reader.

Both un and uno can mean a, but they are used in different situations.

Use un before most masculine singular nouns:

• un libro
• un ragazzo
• un regalo

Use uno before masculine singular nouns beginning with:

• s + consonant: uno studente
• z: uno zaino
• ps, gn, and some others

Since libro begins with a normal l, the correct form is un libro.

In regalo is a common Italian expression meaning as a gift or for free as a gift.

So:

• ricevere un libro in regalo = to receive a book as a gift

It is an idiomatic phrase, and very common in everyday Italian.

You may also see it in expressions like:

• Ti do questo in regalo. = I’m giving this to you as a gift.
• Un campione in regalo. = A free sample / a complimentary sample.

Yes. That is also correct.

Both are natural:

• La figlia riceve un libro in regalo.
• La figlia riceve in regalo un libro.

The difference is mostly one of style and emphasis. In the original sentence, un libro comes first and in regalo explains how it is received. In the other version, in regalo is placed closer to the verb.

No, Italian word order is often more flexible than English word order.

These versions are all possible, depending on emphasis:

• A gennaio la figlia riceve un libro in regalo.
• La figlia riceve un libro in regalo a gennaio.
• In regalo, la figlia riceve un libro a gennaio.
  This is more marked and less neutral.

The original sentence is a very natural, neutral way to say it.

Yes. If the time is not important or is already known, you can leave it out:

• La figlia riceve un libro in regalo.

That simply means The daughter receives a book as a gift.
Adding A gennaio gives a specific time: in January.

Italian usually does not need subject pronouns when the subject is clear.

Here, the subject is already stated as la figlia, so there is no need to add lei.

Italian often works like this:

• La figlia riceve... = The daughter receives...
• Riceve... = She receives... / He receives... / It receives... depending on context

Because verb endings carry information, Italian can often omit pronouns that English must include.

Yes. Riceve is in the present tense.

In Italian, the present tense can mean:

• something happening now
• a regular or habitual action
• sometimes a near-future event, depending on context

So this sentence could mean:

• In January, the daughter receives a book as a gift
  or
• Every January, the daughter gets a book as a gift, if the wider context suggests a repeated habit.

Yes, sometimes it can. In advertisements or shops, in regalo often means something is given free as a bonus:

• una penna in regalo = a free pen included as a gift

In your sentence, though, the natural meaning is simply as a gift.

You would make both the noun and the verb plural:

• A gennaio le figlie ricevono un libro in regalo.

Changes:

• la figlia → le figlie
• riceve → ricevono

So:

• le figlie = the daughters
• ricevono = receive

You would use the indefinite article una:

• A gennaio una figlia riceve un libro in regalo.

That means a daughter, not the daughter.

So the article changes depending on whether the person is specific:

• la figlia = the daughter
• una figlia = a daughter