Mój brat już nie mieszka za granicą.

Because brat is a masculine singular noun and it is the subject of the sentence, so it is in the nominative case.

• mój = my
• brat = brother

The adjective/pronoun mój has to agree with the noun it describes:

• mój brat = my brother
• moja siostra = my sister
• moje dziecko = my child

So mój is simply the correct nominative masculine singular form here.

Polish does not use articles like English a/an and the.

So:

• brat can mean a brother, the brother, or just brother
• mój brat naturally means my brother

The context tells you whether English should use a, the, or nothing.

In this sentence, już nie means no longer or not anymore.

So:

• już by itself often means already
• but już nie together usually means no longer / not anymore

Examples:

• Już mieszkam w Warszawie. = I already live in Warsaw.
• Już nie mieszkam w Warszawie. = I no longer live in Warsaw.

So in your sentence:

• Mój brat już nie mieszka za granicą.
• = My brother no longer lives abroad.

In Polish, nie normally goes directly before the verb to make it negative.

• mieszka = he/she lives
• nie mieszka = he/she does not live

So:

• Mój brat mieszka za granicą. = My brother lives abroad.
• Mój brat nie mieszka za granicą. = My brother does not live abroad.

When you add już, you get:

• Mój brat już nie mieszka za granicą. = My brother no longer lives abroad.

Mieszka is the 3rd person singular present tense form of the verb mieszkać (to live, in the sense of to reside).

Conjugation of mieszkać in the present tense:

• ja mieszkam = I live
• ty mieszkasz = you live
• on/ona/ono mieszka = he/she/it lives
• my mieszkamy = we live
• wy mieszkacie = you all live
• oni/one mieszkają = they live

Since mój brat = my brother, the correct form is mieszka.

Not quite. Mieszkać usually means to live somewhere / to reside.

So it fits well with places:

• Mieszkam w Polsce. = I live in Poland.
• Mieszkają w Krakowie. = They live in Kraków.

If you mean to live as in to be alive, Polish usually uses żyć.

Compare:

• Mój brat mieszka za granicą. = My brother lives abroad.
• Mój brat żyje. = My brother is alive.

In your sentence, mieszka is definitely about where he lives.

Because the natural Polish expression for abroad is za granicą.

Literally, it comes from:

• za = beyond / outside
• granica = border

So za granicą literally suggests beyond the border, but in normal usage it simply means abroad.

English speakers often want to build something like in abroad, but Polish does not do that. The standard expression is:

• mieszkać za granicą = to live abroad
• pracować za granicą = to work abroad
• studiować za granicą = to study abroad

Because after za in this expression, Polish uses the instrumental case.

Base form:

• granica = border

Instrumental singular:

• granicą

So:

• za granicą = abroad / beyond the border

This is one of those common prepositional patterns you gradually learn as a chunk. It is best to memorize za granicą as a whole expression.

Yes, Polish word order is more flexible than English, though some versions sound more natural than others.

The most neutral version is:

• Mój brat już nie mieszka za granicą.

You may also hear:

• Mój brat nie mieszka już za granicą.
• Już mój brat nie mieszka za granicą.
  (possible, but more marked and context-dependent)

The original sentence sounds natural and neutral. Putting już before nie mieszka clearly gives the sense of no longer.

Yes, but the meaning changes.

• Mój brat nie mieszka za granicą. = My brother does not live abroad.
• Mój brat już nie mieszka za granicą. = My brother no longer lives abroad.

Without już, it is just a simple negation.
With już nie, it implies that he used to live abroad before.

Yes. Polish usually does not need subject pronouns like he, she, or I when the verb form already shows the person.

So instead of saying:

• On już nie mieszka za granicą.

you can simply say:

• Już nie mieszka za granicą.

And if you want to specify who, you can use a noun phrase:

• Mój brat już nie mieszka za granicą.

Using on is possible, but often unnecessary unless you want emphasis or contrast.

A rough pronunciation guide:

• Mój ≈ mooy
• brat ≈ braht
• już ≈ yoozh
• nie ≈ nye
• mieszka ≈ MYESH-kah
• za ≈ zah
• granicą ≈ grah-NEE-tsom
  (with a nasal ending)

A few useful notes:

• ó sounds like u
• rz / ż sound like the s in measure
• nie before a consonant-heavy verb like mieszka sounds like nye
• ą is nasal; at the end of a word it is often approximated by learners as om

The most natural translations are:

• My brother no longer lives abroad.
• My brother doesn’t live abroad anymore.

Both are good.
If you want to reflect the Polish structure most closely, no longer is especially neat because it matches już nie very well.