Learn Hardcore Turkish: Çocukların geç saatlere kadar dışarıda kalmasına izin yok. - Children are not allowed to stay outside until late.

Because in the “permission” pattern with izin var/yok/vermek, the doer (subject) of the action is marked with the genitive and the action is turned into a noun. The core pattern is:

X-in Y-mA-sI-na izin var/yok = There is/isn’t permission for X’s doing Y.

Here: çocuk-lar-ın (children-GEN) + … kal-ma-sı-na (to their staying) + izin yok (no permission).

What exactly is kalmasına made of? What does each suffix do?

Breakdown:

kal- = stay
-ma = verbal noun/nominalizer (“staying”)
-sı = 3rd person possessive (“his/her/its/their staying,” structurally required here)
-na = dative case “to/for”

Note: The -n- is a buffer letter that appears after 3rd person possessive -sı when another vowel-starting suffix (like -a/-e) follows. So: kal-ma-sı-na = “to (their) staying.”

Why kalmasına and not kalmaya?

Because you explicitly name the subject in the genitive (çocukların). When the agent is in the genitive, the nominalized verb must carry the appropriate possessive (-sı here): çocukların … kalma-sı-na.

With a stated agent: Çocukların dışarıda kalmasına izin yok. (Correct)
Without stating who: In everyday speech you may see “-mAya izin yok” (e.g., Geç saatlere kadar dışarıda kalmaya izin yok). In more careful/formal style, with transitive verbs people often use a passive + -mesine (e.g., Fotoğraf çekilmesine izin yok). But as soon as you add a genitive subject like çocukların, you need -ması/-mesi.

Can we say kalmalarına instead of kalmasına? What’s the difference?

Both are grammatical:

kalmasına = 3rd person singular possessed form (default, very common for generic rules)
kalmalarına = 3rd person plural possessed form

Nuance: The plural possessed form can highlight that multiple individuals are involved (each child’s staying). The singular possessed form treats the action more generically. In most rule-like statements, the singular possessed form (kalmasına) is preferred, but kalmalarına is also fine.

Why is geç saatlere plural and in the dative? Could it be singular?

It’s in the dative because the postposition kadar (“until, up to”) typically takes a dative-marked phrase: saat-ler-e kadar.
It’s plural because the idiomatic expression is “geç saatler” (“late hours”).
Use singular with a specific clock time: Saat on bire kadar (“until 11 o’clock”), not “geç saate kadar.”

What does kadar do here? Are there synonyms?

Kadar is a postposition meaning “until, up to” (with time/limit meanings) and it follows a dative phrase. Synonyms with roughly the same function are dek and değin:

geç saatlere kadar = geç saatlere dek / geç saatlere değin

Why is it dışarıda and not dışarı?

Kalmak (“to stay/remain”) takes a locative-marked place: X-da/de kalmak. So:

dışarıda kalmak = to stay outside (correct for location)
dışarı kalmak tends to mean “to be left out/excluded,” not simply “to remain outside.” Use dışarı(ya) with motion verbs (çıkmak dışarıya “to go out”).

Can I move the adverbials around? Is the word order fixed?

It’s flexible. For example:

Çocukların dışarıda geç saatlere kadar kalmasına izin yok.
Çocukların geç saatlere kadar dışarıda kalmasına izin yok. (your sentence) Both are natural. Keep kadar immediately after the time phrase it belongs to (geç saatlere).

How else can I express this meaning (with yasak or a verb)?

Common rephrasings:

Çocukların geç saatlere kadar dışarıda kalması yasak. (…is forbidden.)
Okul, çocukların … kalmasına izin vermiyor. (“The school doesn’t allow …”)
Çocuklar … kalamaz. (deontic modal; “Children may not/cannot stay …”)

Why yok and not değil?

Yok is the existential negative of var (“there is/are”). İzin yok = “there is no permission.” Değil negates identity/equative statements (“X is not Y”), which is not what’s happening here.

What’s the difference between izin yok and izni yok?

izin yok = “there is no permission” (impersonal/existential).
izni yok = “his/her/its permission is not there; he/she/it doesn’t have permission.” It requires a possessor: Ali’nin izni yok (“Ali doesn’t have permission”). In our sentence we’re stating that permission (for that action) does not exist in general.

How would I ask “Are children allowed to stay out late?”

Çocukların geç saatlere kadar dışarıda kalmasına izin var mı? Also possible:
Çocukların … kalmasına izin veriliyor mu? (passive/impersonal “is permission being given?”)

If I want to make it specific (“These children are not allowed…”), how do I say it?

Add a determiner:

Bu çocukların geç saatlere kadar dışarıda kalmasına izin yok.
Bizim çocukların … izin yok. (our children)

Is dışarda acceptable instead of dışarıda?

Yes. Dışarda is a common colloquial/short form; dışarıda is the standard/written form. Meaning is the same.

Any quick pronunciation tips for tricky vowels/consonants here?

The dotless ı in çocukların, dışarıda, kalmasına, izin is a back unrounded vowel [ɯ], not like English “i.”
ç = “ch” in “church”; ş = “sh.”
A natural stress pattern is often toward the end of words; yok tends to carry strong stress in this sentence.