Mój brat chce pracować na etat w tej firmie.

Because mój brat is the subject of the sentence, so it stays in the nominative case.

• mój = my
• brat = brother

In Polish, the subject is normally in the nominative:

• Mój brat chce pracować. = My brother wants to work.

Forms like mojego brata are used in other cases, for example:

• Nie widzę mojego brata. = I don’t see my brother.
  Here, mojego brata is in the genitive/accusative-type form used after negation.

Yes. Mój is a possessive adjective, and it must agree with the noun it describes.

Since brat is:

• masculine
• singular
• nominative

the correct form is mój.

Compare:

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

So mój is not a fixed form meaning only my; it changes depending on the noun.

Because after chcieć (to want), Polish usually uses another verb in the infinitive, just like English often does:

• chcę pracować = I want to work
• chce pracować = he/she wants to work

So the structure is:

• [conjugated form of chcieć] + [infinitive]

Examples:

• Chcę spać. = I want to sleep.
• Ona chce kupić samochód. = She wants to buy a car.

You do not conjugate the second verb here:

• correct: Mój brat chce pracować
• not: Mój brat chce pracuje

Chce is the 3rd person singular form of chcieć (to want), used with he, she, or it.

Since mój brat = my brother, we need he wants, so:

• ja chcę = I want
• ty chcesz = you want
• on/ona/ono chce = he/she/it wants

So:

• Mój brat chce... = My brother wants...

Pracować means to work in a general, ongoing sense.

It is the normal verb when talking about:

• having a job
• working somewhere
• working in general

Examples:

• Pracuję w szkole. = I work at a school.
• Chce pracować w tej firmie. = He wants to work in this company.

It is also the imperfective form, which fits here because the sentence is about employment or ongoing work, not a single completed action.

If you used a perfective verb, it would suggest a completed result, which is not the point here.

Na etat means working as a regular employee, usually understood as full-time employment or a salaried/stable position, depending on context.

In everyday Polish, etat is strongly connected with the idea of an official job position rather than casual or freelance work.

So:

• pracować na etat = to work full-time / to be employed in a regular position

Related phrases:

• na pełen etat = full-time
• na pół etatu = part-time
  literally for half a position
• mieć etat = to have a regular job position

This phrase is somewhat idiomatic, so it is best learned as a chunk:

• pracować na etat

Because after w meaning in for location, Polish uses the locative case.

The base form is:

• ta firma = this company

But after w for location, it changes to:

• w tej firmie = in this company

Both words change:

• ta → tej
• firma → firmie

This is very common in Polish:

• w domu = in the house
• w szkole = at school / in the school
• w tej firmie = in this company

Because tej is the feminine singular locative form of ten/ta/to (this).

Since firma is a feminine noun, and after w we need the locative, the correct form is:

• nominative: ta firma = this company
• locative: w tej firmie = in this company

This pattern appears often with feminine nouns:

• ta książka → w tej książce
• ta szafa → w tej szafie
• ta firma → w tej firmie

The word order can change, because Polish is more flexible than English. However, the given order is the most neutral and natural.

Neutral order:

• Mój brat chce pracować na etat w tej firmie.

Possible alternatives:

• W tej firmie mój brat chce pracować na etat.
  This emphasizes in this company.
• Na etat mój brat chce pracować w tej firmie.
  This emphasizes full-time/as a regular employee.

Even though the order can move around, the original sentence sounds the most straightforward for a learner.

Chce is a little tricky for English speakers.

It is pronounced approximately like h-tseh, with:

• ch sounding like the ch in Scottish loch or German Bach
• c sounding like ts
• e like eh

So chce is roughly:

• htseh or kh-tseh

Many learners find the consonant cluster difficult at first, but that is normal.

In normal speech, w often sounds like f before a voiceless consonant, and tej is pronounced roughly like tey.

So:

• w tej firmie sounds approximately like f tey FEER-mye

A rough guide:

• w → f here
• tej → tey
• firmie → FEER-mye

This kind of sound change is very common in Polish pronunciation and does not affect spelling.

Yes, but the meaning and style change a bit.

• firma = the most general, everyday word for company
• spółka = company/corporation, often more formal or legal
• przedsiębiorstwo = enterprise/business, more formal

So in everyday speech:

• w tej firmie is the most natural choice

For most learners, firma is the best word to use unless the context is specifically legal or formal.