Rozpoznaję fragment i kojarzę tę ciekawostkę z piosenką.

Because both verbs are in the 1st person singular present tense, so they mean I recognize and I associate.

• rozpoznaję = I recognize
• kojarzę = I associate / connect

In Polish, the subject pronoun ja (I) is usually omitted, because the verb ending already shows who is doing the action.

So:

• Rozpoznaję = I recognize
• Ja rozpoznaję = also possible, but usually used only for emphasis

Polish often drops subject pronouns when they are not needed.

The ending -ę already tells you the subject is I. So rozpoznaję and kojarzę automatically include I.

Using ja is possible, but it normally adds emphasis, contrast, or clarification:

• Rozpoznaję fragment = I recognize the fragment
• Ja rozpoznaję fragment, ale on nie = I recognize the fragment, but he doesn’t

Because the phrase is the direct object of kojarzę, so it must be in the accusative case.

The base form is:

• ta ciekawostka = this fun fact / trivia item / curiosity
  nominative

But after a verb like kojarzyć when it means to associate something, you need the accusative:

• tę ciekawostkę = accusative

So the changes are:

• ta → tę
• ciekawostka → ciekawostkę

This is a very common pattern for feminine nouns ending in -a:

• widzę kobietę from kobieta
• mam książkę from książka
• kojarzę ciekawostkę from ciekawostka

Because the preposition z here means with / associated with, and in that meaning it takes the instrumental case.

So:

• base form: piosenka
• after z meaning with: z piosenką

This is why the ending changes from -a to -ą.

Compare:

• z kolegą = with a friend
• z książką = with a book
• z piosenką = with a song

No. z can take different cases depending on the meaning.

1. z + instrumental = with

   • z piosenką = with a song
   • z mamą = with mum

2. z + genitive = from / out of

   • z domu = from the house
   • z Polski = from Poland

In your sentence, the meaning is association with a song, so it uses instrumental:

• kojarzę tę ciekawostkę z piosenką

Here kojarzę means something like:

• I associate
• I connect
• I link in my mind

It is a very common Polish verb and can be used in a few related ways.

Examples:

• Kojarzę go z pracy. = I know / recognize him from work.
• Kojarzę tę nazwę. = That name rings a bell / I recognize that name.
• Kojarzę tę ciekawostkę z piosenką. = I associate this fun fact with a song.

So it often involves a mental connection, not necessarily a strong factual statement.

These verbs are related to knowing/recognizing, but they are not the same.

• rozpoznaję = I recognize
• znam = I know / am familiar with
• poznaję = I am getting to know / I meet / I learn to recognize

So in this sentence:

• Rozpoznaję fragment means I recognize the fragment
  → I hear or see it and identify it.

By contrast:

• Znam ten fragment = I know this fragment
• Poznaję ten fragment would usually mean I am starting to recognize this fragment or would sound less natural in many contexts

A useful contrast:

• rozpoznać kogoś = to recognize someone
• poznać kogoś = to meet someone / get to know someone

Fragment is a masculine inanimate noun, and in the singular its accusative form is usually the same as its nominative form.

So:

• nominative: fragment
• accusative: fragment

That is why after rozpoznaję you still see fragment, with no visible change.

Compare:

• To jest fragment. = nominative
• Rozpoznaję fragment. = accusative, but same form

This is normal for many masculine inanimate nouns in Polish.

The sentence uses imperfective verbs:

• rozpoznaję
• kojarzę

That is natural when talking about a current mental action, a general ability, or what happens as you process something.

If you used the perfective forms, the meaning would shift toward a completed result, and in Polish the perfective form does not normally have a present meaning. It usually refers to the future.

For example:

• rozpoznam = I will recognize
• skojarzę = I will associate / make the connection

So:

• Rozpoznaję fragment i kojarzę tę ciekawostkę z piosenką.
  = present, ongoing/current mental recognition

But:

• Rozpoznam fragment i skojarzę tę ciekawostkę z piosenką.
  = more like I will recognize the fragment and make the association

Yes. Polish word order is more flexible than English, because the case endings show grammatical roles.

The sentence as given is neutral and natural:

• Rozpoznaję fragment i kojarzę tę ciekawostkę z piosenką.

But other orders are also possible, depending on emphasis:

• Tę ciekawostkę kojarzę z piosenką.
  Emphasizes this fun fact
• Z piosenką kojarzę tę ciekawostkę.
  Emphasizes with a song
• Fragment rozpoznaję...
  Possible, but sounds more marked or contrastive

So the original order is a good default, especially for learners.

Not exactly. Although ciekawostka is related to the idea of something curious, in everyday Polish it often means:

• fun fact
• interesting tidbit
• trivia item
• little interesting piece of information

So in many contexts, translating it literally as curiosity sounds unnatural in English.

For example:

• Mam ciekawostkę o Warszawie.
  = I have a fun fact about Warsaw.

In your sentence, tę ciekawostkę is best understood as this interesting fact / this bit of trivia.

Piosenka means song, but it often sounds a little more like:

• song in an everyday/popular sense
• sometimes slightly lighter or more informal than utwór (musical piece / track / work)

So:

• piosenka = ordinary everyday word for song
• utwór = broader, sometimes more formal or technical

In your sentence, piosenka is completely natural.

A few parts may stand out to English speakers:

• rz in kojarzę is pronounced like ż
• ci in ciekawostkę sounds like a soft ch / ć-type sound before the vowel
• ę is a nasal vowel, though in normal speech its pronunciation can vary depending on position
• ą in piosenką is also nasal

A rough learner-friendly approximation:

• Rozpoznaję ≈ roz-poz-NA-yeh
• kojarzę ≈ ko-YA-zheh
• ciekawostkę ≈ che-ka-VOST-keh
• piosenką ≈ pyo-SEN-kaw̃

That said, the exact pronunciation of ę and ą depends on what sounds come after them, so it is best to learn them by listening to native speech rather than relying only on English-style approximations.

Yes, if you mean with that particular song.

Your original sentence has:

• z piosenką = with a song

If you want to specify that song, you would say:

• z tą piosenką = with that song

Notice the instrumental again:

• ta piosenka → z tą piosenką

So both are correct, but the meaning changes:

• z piosenką = with a song, non-specific
• z tą piosenką = with that specific song