Learn Hardcore Polish: Rano dzieci szybko myją zęby, żeby zdążyć na wschód słońca. - In the morning the children quickly brush their teeth to manage to see the sunrise.

In your sentence, without any extra determiners, it most naturally sounds like a general habit of (some) children, but in a story it could refer to specific, previously mentioned children.

What exactly is rano? Why don’t we say w rano like “in the morning”?

Rano is an adverb meaning in the morning. It already includes the idea of “in”, so you do not add a preposition:

rano – in the morning
wieczorem – in the evening
w nocy – at night (here w is needed, this one is irregular)

You can’t say *w rano; that’s ungrammatical. Another option with the same meaning is:

rankiem – also “in the morning”

Both rano and rankiem are very common.

Is the word order “Rano dzieci szybko myją zęby” fixed, or can I move the words around?

Polish word order is relatively flexible. All of these are grammatically correct:

Rano dzieci szybko myją zęby.
Dzieci rano szybko myją zęby.
Dzieci szybko myją zęby rano.
Szybko rano dzieci myją zęby. (sounds a bit marked/stylised)

The default version Rano dzieci szybko myją zęby puts rano first, making the time (“in the morning”) the background frame of the sentence.
If you move words, you slightly change what’s emphasised, but the basic meaning stays the same. For most everyday use, you can treat these as equivalent and choose the one that sounds most natural to you, especially:

Rano dzieci szybko myją zęby.
Dzieci rano szybko myją zęby.

Why is it myją zęby and not something like myją się zęby or a special verb for “brush”?

In Polish, myć means to wash, and you wash a direct object:

myć ręce – to wash hands
myć włosy – to wash hair
myć zęby – literally “to wash teeth”, but this is how you say to brush your teeth

myć się (with się) means “to wash oneself” in general:

Dzieci myją się. – The children are washing themselves / having a wash.

So:

Dzieci myją zęby. – The children brush their teeth.
Dzieci myją się. – The children are washing (their bodies, faces, etc.).

There is a verb szczotkować zęby (“to brush teeth”), but in everyday speech myć zęby is much more common and natural.

What form and case is zęby? Why isn’t it ząby or zębów?

The basic (dictionary) form is ząb – tooth. The noun is irregular:

singular nominative: ząb – a tooth
plural nominative: zęby – teeth

In your sentence, zęby is the accusative plural (direct object):

(Kto?) dzieci – subject, nominative
(co robią?) myją – verb
(co?) zęby – object, accusative

For this noun, nominative plural = accusative plural, both zęby.

Why is the verb myją (3rd person plural) and not myje? Is dzieci grammatically singular or plural?

Dzieci is grammatically plural, so the verb must be plural:

(to) dziecko myje zęby – The child brushes its teeth.
(te) dzieci myją zęby – The children brush their teeth.

So:

myje – 3rd person singular
myją – 3rd person plural (used with dzieci)

What does żeby do here, and how is it different from just using żeby + a full clause like żeby dzieci zdążyły?

In this sentence żeby introduces a purpose:

…myją zęby, żeby zdążyć na wschód słońca.
→ “…they brush their teeth in order to make it for the sunrise.”

Two common patterns with żeby:

żeby
- past/future form (finite verb)
- Dzieci szybko myją zęby, żeby zdążyły na wschód słońca.
  Literally: “[so that] the children would/so that they manage…”
żeby
- infinitive (like your sentence)
- Dzieci szybko myją zęby, żeby zdążyć na wschód słońca.
  “[in order] to make it for the sunrise.”

When the subject of both parts is the same (here: dzieci), Polish very often uses żeby + infinitive. It sounds natural and is very common in speech and informal writing.
Żeby + full clause (…żeby dzieci zdążyły) is also correct; it can sound a bit more explicit or formal.

Could I replace żeby with aby here? Is there a difference?

You can say:

Rano dzieci szybko myją zęby, aby zdążyć na wschód słońca.

It’s grammatically fine. The difference:

żeby – the most common, neutral, used everywhere (spoken and written).
aby – more formal, slightly bookish or official in many contexts.

In everyday conversation people overwhelmingly use żeby. In written instructions, essays, or formal texts you’ll see aby more often. In your sentence, żeby is the most natural choice.

What exactly does zdążyć mean, and why is this perfective form used instead of some present-tense form like zdążają?

Zdążyć is a perfective verb meaning roughly:

to manage to be on time / to make it (before a deadline/event)

Examples:

Zdążyliśmy na pociąg. – We managed to catch the train.
Muszę zdążyć przed zachodem słońca. – I must make it before sunset.

Because it’s perfective, it focuses on the result: being in time for the sunrise. That’s exactly what we want here.

There is an imperfective partner zdążać, but:

it’s much less common,
and it’s used more for repeated or ongoing situations (and even then, many speakers prefer other verbs like wyrabiać się, dawać radę, etc.).

For a specific goal like “to be in time for the sunrise this morning”, the perfective zdążyć is the natural choice.

Why is it na wschód słońca? What case is that, and how is it different from o wschodzie słońca?

na wschód słońca consists of:

na
- wschód – accusative (direction / target)
słońca – genitive of słońce (sun); literally “rising of the sun”.

So literally: “to the rising of the sun” → for the sunrise.

na + accusative is used for aiming at or making it to an event:

zdążyć na pociąg – catch the train
iść na koncert – go to a concert
przyjść na spotkanie – come to a meeting
zdążyć na wschód słońca – make it for the sunrise

o wschodzie słońca uses:

o
- wschodzie – locative → “at the time of the sunrise”

So:

Zdążyli na wschód słońca. – They made it for the sunrise (they arrived before/just in time).
Obudzili się o wschodzie słońca. – They woke up at sunrise.

Both are correct, but the preposition + case combination changes the meaning slightly.

Why is there a comma before żeby?

In Polish, you put a comma before most subordinating conjunctions, including żeby, że, gdy, ponieważ, etc.

Here:

Rano dzieci szybko myją zęby, – main clause
żeby zdążyć na wschód słońca. – purpose clause introduced by żeby

Because żeby zdążyć na wschód słońca depends on the main clause and explains the purpose, a comma is required before żeby.

What kind of word is szybko, and can I move it around the sentence?

Szybko is an adverb meaning quickly / fast. It modifies the verb myją.

You can move it to a few different positions:

Rano dzieci szybko myją zęby. – default, very natural.
Rano dzieci myją zęby szybko. – also fine; slightly more emphasis on how they brush.
Dzieci szybko myją zęby rano. – ok, but sounds a bit less neutral.

In Polish, adverbs like szybko, wolno, dokładnie are quite flexible in position, but putting szybko right before the verb is the most common pattern.

Is there anything special about dzieci in terms of gender?

Dzieci is grammatically plural, and its gender class is treated like non‑masculine personal plural (the same group as kobiety – women, koty – cats, etc.).

For verb agreement, you just need to remember:

dziecko – singular → myje
dzieci – plural → myją

For adjectives and past tense you would also use non-masculine-personal plural endings:

małe dzieci – small children
Dzieci były zmęczone. – The children were tired.