Po pracy jestem trochę zmęczony.

Because po meaning after normally requires the locative case in Polish.

The basic form is praca = work.
After po, it changes to pracy:

• praca → base form
• po pracy → after work

So this is a case change, not a different word.

It is the locative singular.

With po in the sense of after, Polish uses the locative:

• po szkole = after school
• po obiedzie = after dinner
• po pracy = after work

A useful thing to remember is that pracy looks the same as the genitive singular form, but in this sentence its function is locative, because that is what po requires.

Because Polish usually does not need the subject pronoun when the verb already shows the person.

jestem means I am, so ja is unnecessary:

• Jestem trochę zmęczony. = I am a bit tired.
• Ja jestem trochę zmęczony. = I am a bit tired, with extra emphasis on I

So the normal neutral version omits ja.

In Polish, states like being tired, happy, hungry, ready are usually expressed with to be + adjective, just like in English.

So:

• jestem zmęczony = I am tired

You do not normally say mam zmęczenie for this meaning. That would sound unnatural in everyday speech.

This pattern is very common:

• jestem głodny = I am hungry
• jestem gotowy = I am ready
• jestem zmęczony = I am tired

Trochę means a little, a bit, or somewhat.

So jestem trochę zmęczony means:

• I am a bit tired
• I am somewhat tired

It softens the statement. Without it:

• Po pracy jestem zmęczony. = After work I am tired.

With trochę:

• Po pracy jestem trochę zmęczony. = After work I am a little tired.

Because trochę modifies the adjective zmęczony and naturally comes before it.

So:

• jestem trochę zmęczony = I am a bit tired

That is the most natural word order.

Polish word order is fairly flexible, but this version is the most neutral and standard.

Yes. Zmęczony is an adjective, so it changes for gender and number.

For one person:

• zmęczony = masculine
• zmęczona = feminine
• zmęczone = neuter

Examples:

• If a man says it: Po pracy jestem trochę zmęczony.
• If a woman says it: Po pracy jestem trochę zmęczona.

For plural, the form changes too, for example:

• Jesteśmy trochę zmęczeni. = We are a bit tired.

Yes, Polish word order is flexible, but different orders can sound more or less natural depending on emphasis.

The neutral version is:

• Po pracy jestem trochę zmęczony.

You could also hear:

• Jestem trochę zmęczony po pracy.
• Trochę jestem zmęczony po pracy. — possible, but less neutral

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

Usually it just means after work in a general sense. It often implies the time after finishing work, and depending on context it may also suggest when I get home from work.

So it is broad, just like English after work.

If you wanted to be more specific, Polish could say things like:

• kiedy wracam z pracy = when I come back from work
• po powrocie z pracy = after returning from work

But po pracy is the normal simple expression.

A rough guide:

• pracy ≈ PRA-tsih
• zmęczony ≈ zmeh-CHO-nih

A few useful sound notes:

• c in pracy sounds like ts
• cz in zmęczony sounds like English ch, but harder
• ę in zmęczony is a nasal vowel; before cz it is often pronounced approximately like em/en, so learners often hear something like zmen-cho-ny
• stress in Polish usually falls on the second-to-last syllable:
  • PRA-cy
  • zmę-CZO-ny

Yes, but it means something different.

• Jestem po pracy. = I’m done with work / I’m after work / I’ve finished work.
• Po pracy jestem trochę zmęczony. = After work I’m a bit tired.

So jestem po pracy describes your situation in relation to work, while the original sentence describes how you feel after work.

It is neutral and natural in both everyday speech and standard written Polish.

It is not especially formal and not especially casual. You can use it in normal conversation without any problem.

In standard Polish, you normally keep jestem here.

• Po pracy jestem trochę zmęczony. = correct, complete sentence

In very casual speech, people sometimes drop parts of a sentence if the meaning is obvious, but for learners the full version with jestem is the right one to use.

So yes, omission may happen colloquially, but don’t treat it as the standard pattern.