Dzieci, umyjcie zęby: pasta do zębów jest przy umywalce, a szczoteczka leży obok kubka.

Yes. Dzieci is being used in direct address, like Children, ... in English, so Polish puts a comma after it.

A useful detail: in the plural, dzieci has the same form for both nominative and vocative, so even though it is functioning like a vocative here, it still looks like dzieci.

The ending -cie shows that this is an imperative addressed to you all.

So:

• umyj = wash! to one person
• umyjcie = wash! to more than one person

Since the sentence starts with Dzieci, the speaker is clearly talking to several children.

This is about aspect, which is very important in Polish verbs.

• myć is imperfective
• umyć is perfective

So:

• myjcie zęby means something like brush/wash your teeth in a general or ongoing sense
• umyjcie zęby means wash/brush your teeth and get it done

In a one-time instruction to children, umyjcie sounds very natural because it focuses on the completed action.

Because zęby is the direct object of the verb, so it is in the accusative plural.

The noun is:

• singular: ząb
• plural nominative: zęby
• plural accusative: zęby

For many masculine inanimate nouns in Polish, the plural accusative looks the same as the plural nominative, which is why you see zęby here.

Polish often leaves out words like your when the owner is obvious from context, especially with body parts.

So umyjcie zęby naturally means brush your teeth.

If you said umyjcie swoje zęby, it would be grammatical, but usually less natural unless you want emphasis, contrast, or extra clarity.

Here do does not mean to in the direction sense. It means something more like for or used for.

So:

• pasta do zębów = toothpaste
• literally: paste for teeth

After do, Polish uses the genitive, so zęby changes to zębów.

Because the preposition przy requires the locative case.

The dictionary form is:

• umywalka = sink / washbasin

After przy, it becomes:

• przy umywalce = by the sink / at the sink

So this is a normal case change caused by the preposition.

Because obok takes the genitive case.

The base form is:

• kubek = cup / mug

After obok, it becomes:

• obok kubka = next to the cup

So this is another example of a preposition forcing a case change.

Jest is the general verb to be, so jest przy umywalce simply means is by the sink.

Leży comes from leżeć, meaning to lie. It suggests that the object is physically resting somewhere.

So:

• pasta do zębów jest przy umywalce = the toothpaste is by the sink
• szczoteczka leży obok kubka = the toothbrush is lying next to the cup

In English we often just say is, but Polish often uses more specific location verbs like leży, stoi, or wisi.

Both can often translate as and, but they are not always used the same way.

• i usually just adds one thing to another
• a often links two related pieces of information with a slight contrast or shift of focus

Here, the sentence gives two separate location facts:

• the toothpaste is by the sink
• the toothbrush is next to the cup

So a sounds very natural: it moves from one item to the other.

The colon introduces an explanation or follow-up detail.

The first part gives the instruction:

• Dzieci, umyjcie zęby

Then the colon introduces helpful information about where the things are:

• the toothpaste is by the sink, and the toothbrush is next to the cup

So the punctuation works much like an English colon would.

Because the context already makes it clear that this is a toothbrush.

• szczoteczka = small brush / toothbrush depending on context
• szczoteczka do zębów = toothbrush explicitly

Since the sentence already mentions teeth and toothpaste, Polish can naturally shorten it to just szczoteczka.