Djeca trče po dvorištu nakon škole.

Djeca is the nominative plural form used for the subject of the sentence (The children).

• djeca = nominative (subject) plural
• djece = genitive (often after negation, quantities, “of the children,” etc.)
• djeci = dative/locative (to/for the children; about/at the children depending on context)

Since the children are doing the action (trče), you use nominative: Djeca trče...

Meaning-wise it’s plural (children), and it takes a plural verb: trče (they run).
Croatian treats djeca as a plural subject, so agreement is plural:

• Djeca trče. (not Djeca trči.)

trče means (they) run / are running (present tense).
The dictionary form (infinitive) is trčati (to run).

Croatian present tense often covers both English run and are running; context decides.

Both can be possible but they emphasize different ideas:

• u dvorištu = in the yard (location; being inside that area)
• po dvorištu = around/all over the yard (movement spread across the area)

With trče (running), po commonly highlights moving around within the space, not just being located there.

dvorištu is locative singular (same form as dative singular for many nouns). It’s used because:

• the preposition po (in this meaning of movement within an area) takes the locative.

So: po + locative → po dvorištu.

The base (dictionary) form is dvorište (courtyard/yard).
It’s neuter. Typical neuter pattern:

• nominative: dvorište
• locative: u/po dvorištu

Not always. po can govern different cases depending on meaning:

• po + locative: movement/distribution within an area (po dvorištu = around the yard), or “according to” (po pravilima = according to the rules)
• po + accusative: direction/goal in some uses, especially with amounts or time spans, e.g. po jedan (one each), po cijeli dan (all day long)

In your sentence, it’s the common po + locative “around” use.

Both are correct and very common:

• nakon škole = after school (neutral, slightly more formal)
• poslije škole = after school (often feels more conversational in many regions)

They both take the genitive: škole.

škole is genitive singular of škola (school).
Prepositions like nakon and poslije require the genitive, so:

• nakon + genitive → nakon škole
• poslije + genitive → poslije škole

No, because the case would be wrong:

• poslije školi → školi is dative/locative, but poslije needs genitive
• nakon škola → škola in nominative plural doesn’t fit; you need genitive singular here: škole

Correct: poslije škole / nakon škole.

The given order is very natural: Djeca trče po dvorištu nakon škole.
Croatian word order is flexible, but changes emphasis:

• Nakon škole djeca trče po dvorištu. (emphasizes after school)
• Djeca nakon škole trče po dvorištu. (also fine)
• Po dvorištu djeca trče nakon škole. (emphasizes around the yard)

Most versions are grammatical; the default is usually “subject + verb + (place) + (time).”

It depends on context:

• dvorište can be a yard (e.g., by a house)
• or a school courtyard/playground area

The Croatian word is broader than either English term; you pick the English word that matches the setting.