Половина гостей уже дома.

Because половина requires the following noun to be in the genitive.

• половина гостей = half of the guests
• гостей is the genitive plural form of гости

This is very common in Russian with words that express quantity:

• много гостей = many guests
• часть гостей = part of the guests
• половина гостей = half of the guests

So here, половина is the main noun, and гостей depends on it.

Yes, grammatically половина is the subject.

Even though the sentence is about many people, Russian treats половина as the head noun:

• Половина = half

So the structure is literally something like:

• Half of the guests already at home

That is why, in other tenses, agreement often follows половина, which is singular feminine:

• Половина гостей уже была дома. = Half of the guests were already home.

In Russian, the verb быть (to be) is normally omitted in the present tense in simple statements.

So:

• Половина гостей уже дома.

literally has no present-tense are, but it naturally means:

• Half of the guests are already home.

Compare:

• Он дома. = He is home.
• Мы готовы. = We are ready.

But in the past or future, быть appears:

• Половина гостей уже была дома. = Half of the guests were already home.
• Половина гостей уже будет дома. = Half of the guests will already be home.

Because дома means at home, while домой means to home / homeward.

Here the sentence describes a location/state, not movement:

• дома = at home
• домой = (going) home

Compare:

• Половина гостей уже дома. = Half of the guests are already home.
• Половина гостей уже идёт домой. = Half of the guests are already going home.

This is a very important distinction in Russian.

Because дома is a fixed adverb-like word meaning at home.

So:

• дома = at home
• в доме = in the house / in the building

These are not the same.

For example:

• Я дома. = I am at home.
• Я в доме. = I am in the house.

The first is about being at one’s home; the second is about physical location inside a building.

Yes. Written without stress marks, дома can mean either:

• до́ма = at home
• дома́ = houses

In your sentence, it means до́ма = at home.

The context makes that clear:

• Половина гостей уже дома.
  Here it obviously means at home, not houses.

This stress difference is useful to know:

• до́ма = at home
• дома́ = houses

Уже means already.

In this sentence, it tells us that the situation has happened earlier than expected or is true by now:

• Половина гостей уже дома. = Half of the guests are already home.

Its position is fairly natural here, before дома, but Russian word order is flexible. Different placements can slightly change emphasis:

• Половина гостей уже дома. = neutral
• Уже половина гостей дома. = emphasis on the fact that the number has already reached half
• Половина гостей дома уже. = possible, but less neutral in everyday speech

So the original sentence is a very natural standard way to say it.

Usually, the most natural form is simply:

• половина гостей

Russian normally does not need a separate word for English of in this structure.

So:

• половина гостей = natural, standard
• половина из гостей = possible in some contexts, but more marked and less neutral

Using из can sound like you are explicitly selecting from a known group, almost like half out of the guests. For a normal sentence, половина гостей is best.

The safest standard agreement is singular, because половина is a singular noun:

• Половина гостей уже была дома.

However, with nouns like половина, часть, большинство, Russian sometimes also uses plural agreement, especially when thinking about the people as individuals:

• Половина гостей уже были дома.

So both can occur, but they feel slightly different:

• была = focuses more on half as a quantity/unit
• были = focuses more on the guests themselves

For learners, singular agreement is the safest default in careful Russian.

Because половина itself is a singular noun meaning half.

It does not matter that the people inside that group are plural. Grammatically, the word половина is one thing:

• эта половина = this half

So Russian grammar sees:

• половина = singular feminine noun
• гостей = genitive plural dependent noun

That is why the sentence starts with a singular noun even though the meaning involves multiple people.

Yes. Russian word order is flexible, but changing it changes emphasis, not the core meaning.

For example:

• Половина гостей уже дома.
  Neutral: Half of the guests are already home.

• Уже половина гостей дома.
  Emphasis on already half

• Дома уже половина гостей.
  Emphasis on at home

The original order is the most neutral and natural if you are just stating the fact.