Moja córka nie lubi hummusu, ale chętnie je ogórek kiszony i chleb.

Because moja córka is the subject of the sentence: my daughter.

• moja = my for a feminine singular noun
• córka = daughter

The subject is in the nominative case, so you get moja córka.

A quick comparison:

• moja córka = my daughter
• mojej córki = of my daughter
• moją córkę = my daughter as a direct object

Here, she is the one doing the action, so nominative is correct.

This is a very common Polish pattern: negation often changes the direct object from accusative to genitive.

Without negation:

• lubi hummus = she likes hummus

With negation:

• nie lubi hummusu = she does not like hummus

So:

• hummus = accusative/nominative-looking form
• hummusu = genitive singular

This is one of the classic examples of the genitive after negation rule in Polish.

Chętnie literally means something like willingly, gladly, or with pleasure.

In everyday English, it often sounds more natural as:

• likes to
• readily
• is happy to

So chętnie je is not just eats. It suggests that she eats it gladly / enjoys eating it.

That is why the contrast works well:

• nie lubi hummusu = she doesn’t like hummus
• ale chętnie je... = but she gladly eats...

Je is the 3rd person singular present tense of jeść = to eat.

Conjugation of jeść in the present:

• jem = I eat
• jesz = you eat
• je = he/she/it eats
• jemy = we eat
• jecie = you all eat
• jedzą = they eat

Since moja córka means my daughter, the verb must be she eats, so Polish uses je.

Because Polish often drops subject pronouns when they are already clear from context or from the verb form.

So instead of saying:

• Moja córka ... ale ona chętnie je...

Polish normally just says:

• Moja córka ... ale chętnie je...

The verb form je already tells you it is he/she/it eats, and the earlier subject moja córka makes it clear that she is meant.

Adding ona is possible, but it would usually add emphasis, like she, specifically.

Because after a positive verb like je, the direct object is usually in the accusative, but for many masculine inanimate nouns, the accusative singular looks exactly like the nominative singular.

That is what happens here:

• ogórek = cucumber
• chleb = bread

As objects, they are still accusative, but their forms do not change:

• je ogórek kiszony
• je chleb

So the case is accusative, even though the spelling looks like nominative.

Because ogórek is masculine inanimate.

In Polish:

• masculine animate nouns often have accusative forms like the genitive
• masculine inanimate nouns usually keep a nominative-looking accusative

So:

• widzę psa = I see a dog
• jem ogórek = I eat a cucumber

That is why:

• ogórek kiszony is correct here
• ogórka kiszonego would sound wrong in this sentence

Also, the adjective must match the noun, so if the noun stays ogórek, the adjective stays kiszony.

Kiszony means pickled or more literally fermented in brine.

It comes from kisić / ukisić = to pickle / ferment.

The form kiszony matches ogórek because:

• ogórek is masculine singular
• so the adjective must also be masculine singular

That gives:

• ogórek kiszony = pickled cucumber

If the noun changed, the adjective would change too:

• kapusta kiszona = pickled/sour cabbage
• ogórki kiszone = pickled cucumbers

Ale means but and gives a clear contrast.

So this sentence says:

• she does not like one thing,
• but she gladly eats something else.

You could sometimes see a in similar sentences, because a can also connect contrasting ideas in Polish. But ale is the more direct, stronger equivalent of English but, so it fits very naturally here.

Yes. Polish word order is fairly flexible, because case endings and verb forms carry a lot of grammatical information.

This sentence is neutral and natural:

• Moja córka nie lubi hummusu, ale chętnie je ogórek kiszony i chleb.

But other orders are possible for emphasis, for example:

• Hummusu moja córka nie lubi, ale chętnie je ogórek kiszony i chleb.
  This emphasizes hummus.

• Moja córka chętnie je ogórek kiszony i chleb, ale nie lubi hummusu.
  This puts the positive part first.

So the original order is not the only possible one, but it is a very normal, neutral choice.

A few useful points:

• moja ≈ MO-ya
• córka ≈ TSOOR-ka
• nie lubi ≈ nyeh LOO-bee
• hummusu ≈ hoo-MOO-soo
• chętnie ≈ roughly KHEN-tnye
• je ≈ yeh
• ogórek ≈ oh-GOO-rek
• kiszony ≈ kee-SHO-nih
• chleb ≈ a rough hlep sound

A few specific pronunciation notes:

• ó is pronounced like u
• je sounds like ye
• rz / ż usually sound like the s in measure
• ch is a rough throaty sound, not English ch as in chair
• Polish stress is usually on the second-to-last syllable:
  • CÓR-ka
  • hum-MU-su
  • ki-SZO-ny
  • o-GÓ-rek