Wszystkie dzieci biegają w parku.

Both mean all, but they must agree in gender with the noun:

• wszyscy – used with masculine personal plural (groups that include at least one male person treated as “men/boys”):
  
  • Wszyscy chłopcy = All (the) boys
• wszystkie – used with:
  • feminine plural nouns: wszystkie dziewczyny (all girls)
  • neuter plural nouns: wszystkie dzieci (all children)
  • non‑personal plurals in general: wszystkie krzesła (all chairs)

Dzieci is grammatically neuter plural, so you must use wszystkie.

Dzieci is the plural form of dziecko (child).

• Singular: dziecko (child)
• Plural: dzieci (children)

It is:

• number: plural
• gender: neuter plural (grammatically)
• case in this sentence: nominative (subject of the verb)

So wszystkie dzieci = all (the) children as the subject.

Two cases appear here:

1. Nominative (mianownik) – subject:

   • Wszystkie dzieci
   • Question: kto? co? (who? what?)
   • Answer: wszystkie dzieci (all children – who is running?)

2. Locative (miejscownik) – after the preposition w indicating location:

   • w parku
   • Question: w kim? w czym? (in whom? in what?)
   • Answer: w parku (in the park – where are they running?)

So:
[NOM] Wszystkie dzieci [VERB] biegają [LOC] w parku.

After w you choose the case according to the meaning:

• w + locative → static location (where something is)
  
  • w parku = in the park (place where they are running)
• w + accusative → movement into (where something is going to)
  
  • iść w park is unusual; with park we normally say iść do parku (go to the park)

So in biegają w parku, the action is happening inside / within the park, so parku is locative singular of park:

• Nominative: park
• Locative: (w) parku

You’re right: parku can be both genitive and locative for park.

You tell them apart by:

1. Preposition:

   • With w meaning “in (a place)” → locative
     • w parku = in the park
   • With do meaning “to” → genitive
     • do parku = to the park

2. Meaning:

   • If it answers “where?” → locative
   • If it answers “of whom/what?”, “to/from where?” → often genitive

Here, w parku answers where (are they running)? → locative.

Polish does not use a continuous tense like English am/is/are + -ing.

• English: They run / They are running
• Polish: both are simply Oni biegają or Dzieci biegają

The present tense of an imperfective verb (like biegać) already covers:

• habitual: They (often) run
• current: They are (now) running

A form like są biegające is grammatically possible but unnatural and rarely used in this meaning. So biegają alone is the normal way to say are running.

They come from different verbs/aspects:

• biegać (imperfective, frequent/indefinite movement)
  
  • biegają – they run / they are running (in general, around, back and forth)
• biec (imperfective but usually single, directed movement)
  
  • biegną – they are running (in a specific direction, in one “go”)

In your sentence:

• Wszystkie dzieci biegają w parku – they’re running around in the park, moving here and there (typical image of children playing).
• Wszystkie dzieci biegną do parku – all the children are running to the park (they’re going there).

So biegają w parku suggests more “running around in the park” than “running in a straight line”.

Biegać (to run, to go running) – present tense:

• ja biegam – I run
• ty biegasz – you run (sg.)
• on/ona/ono biega – he/she/it runs
• my biegamy – we run
• wy biegacie – you run (pl.)
• oni/one biegają – they run

So biegają is:

• person: 3rd
• number: plural
• used with oni or one

Here it agrees with dzieci (neuter plural), which takes the one / biegają pattern.

In Polish, the group dzieci is treated grammatically as neuter plural, non‑personal, even though children are human beings.

That means:

• pronoun: one (not oni)
  
  • Dzieci są małe. One biegają.
• adjective/pronoun agreement: wszystkie dzieci, małe dzieci, one były zmęczone

So the verb form biegają is the same as for one (non‑personal plural), and the quantifier is wszystkie, not wszyscy.

Yes, Polish word order is more flexible than English. These are all correct:

• Wszystkie dzieci biegają w parku.
• W parku wszystkie dzieci biegają.
• W parku biegają wszystkie dzieci.
• Wszystkie dzieci w parku biegają. (a bit marked/emphatic)

The basic neutral order is subject–verb–place:

• Wszystkie dzieci (subject)
• biegają (verb)
• w parku (place)

Moving parts around can add emphasis:

• W parku wszystkie dzieci biegają. – emphasizes in the park
• W parku biegają wszystkie dzieci. – emphasizes that it’s all the children who are running there.

Both w and na can mean in / at / on, but they are used with different types of nouns, often by convention:

• w: closed spaces, interiors, cities, countries, many public buildings
  
  • w parku (in the park)
  • w szkole (at school)
  • w domu (at home)
• na: open surfaces, some institutions, events, and some places by idiom
  
  • na placu zabaw (on the playground)
  • na dworcu (at the station)
  • na uniwersytecie (at university)

With park, the normal phrase is w parku, not na parku.

Yes, mainly in how inclusive it sounds:

• Dzieci biegają w parku.

  • Children are running in the park / Some children are running in the park / Children (in general) run in the park.
  • Could be generic or partial.

• Wszystkie dzieci biegają w parku.

  • All (the) children are running in the park.
  • Explicitly includes every child in the relevant group (e.g. all the children we’re talking about).

So wszystkie adds the idea of all, every one of them.

Approximate pronunciation for an English speaker:

• dzieci – [JEC-ee] but with a soft j and ch:

  • dz – like j in jeans, but a bit harder
  • ie – like ye in yes
  • ć – a soft ch sound (like t in nature in some accents)
  • whole word: DJEH‑chee (one smooth unit)

• biegają – [bye-GAH-yon] (roughly)

  • bie – like bye but shorter, closer to bye
    • eh
  • ga – ga as in gaga
  • ją / ą – nasal vowel, something like on in French bon, but you don’t fully pronounce n
  • very rough: bye‑GAH‑yon (nasalizing the last sound)

Native pronunciation is smoother, but these approximations will make you understood.