Learn Hardcore German: Die Kinder schauen dem Hund im Garten zu. - The children watch the dog in the garden.

sehen = to see (more neutral, just using your eyes)
- Ich sehe den Hund. – I see the dog.
schauen (especially in southern German / Austrian usage) = to look
Often needs a preposition or particle:
- Schau mal! – Look!
- Wir schauen Fernsehen. – We watch TV.
zuschauen = to watch (to observe something that is happening, usually over some time)
- Die Kinder schauen dem Hund zu. – The children watch the dog (e.g. as it runs/plays).

So in this sentence, schauen … zu = zuschauen = to watch, not just accidentally seeing the dog.

Could I also say Die Kinder sehen den Hund im Garten? Does it mean the same?

You can say that, but the meaning is slightly different.

Die Kinder sehen den Hund im Garten.
→ They see the dog in the garden (they notice him visually; no idea how long or what else happens).
Die Kinder schauen dem Hund im Garten zu.
→ They watch the dog in the garden (they are actively observing what he is doing, for some time).

So sehen focuses on the fact of seeing; zuschauen/watching suggests attention and duration.

What case is Die Kinder in, and how do we know?

Die Kinder is nominative plural.

We know this because:

It is the subject of the sentence – the ones who are doing the action of watching.
The definite article die is used for:
- nominative plural (all genders)
- accusative plural (all genders)

Here it must be nominative, because someone has to perform the action schauen. So:

Die Kinder (nominative plural) = the children (subject).

Why is it im Garten and not in dem Garten?

im is just a standard contraction of in dem:

in dem Garten → im Garten

Native speakers almost always use im in everyday speech and writing unless they need special emphasis or style.

Grammatically:

in
- dative for location (where?)
der Garten (masculine, nominative)
dative singular masculine article = dem
→ in dem Garten → contracted to im Garten

Why is Garten also in the dative? Why not accusative?

Because in takes:

dative when it expresses location (where?)
accusative when it expresses direction / movement (where to?)

In this sentence, the dog is in the garden (location, no movement indicated):

im Garten = in dem Garten – in the garden (dative)

If you wanted to express movement into the garden:

Der Hund läuft in den Garten. – The dog runs into the garden. (accusative)

How do I know whether im Garten describes where the children are or where the dog is?

The sentence is slightly ambiguous in theory, but typical interpretation and word order help:

Die Kinder schauen dem Hund im Garten zu.

In German, a phrase like im Garten placed right after dem Hund is usually understood as describing the dog (the thing you’re watching), not the subject. So the most natural reading is:

The children (location unspecified) are watching the dog who is in the garden.

If you really wanted to emphasize that the children are in the garden, you’d more likely say:

Im Garten schauen die Kinder dem Hund zu. – In the garden, the children watch the dog.

Die Kinder im Garten schauen dem Hund zu. – The children in the garden watch the dog. (now Kinder is clearly modified)

Can I change the word order, e.g. Die Kinder schauen im Garten dem Hund zu?

Yes, that is grammatically correct. German word order is relatively flexible as long as:

the conjugated verb stays in the 2nd position in a main clause, and
the separable prefix (zu) stays at the end.

Some possible versions:

Die Kinder schauen dem Hund im Garten zu. (neutral, very natural)
Die Kinder schauen im Garten dem Hund zu. (slight emphasis on in the garden)
Im Garten schauen die Kinder dem Hund zu. (emphasis on the place: In the garden...)

All of these are correct; the differences are mainly in emphasis, not grammar.

What gender is Hund, and what are its main article forms?

Hund is masculine (der Hund).

Key singular forms with the definite article:

Nominative: der Hund – the dog (subject)
Accusative: den Hund – the dog (direct object)
Dative: dem Hund – to/for the dog (indirect object or after certain verbs, like zuschauen)
Genitive: des Hundes

In our sentence, dem Hund is dative singular masculine.

Why do we use an article with Kinder (Die Kinder) instead of just Kinder?

German usually needs an article (or another determiner like meine, diese) with plural nouns, unless you’re making a very general statement or using certain fixed expressions.

Die Kinder spielen. – The children are playing. (a specific group we have in mind)
Kinder spielen gern. – Children like to play. (children in general)

In Die Kinder schauen dem Hund im Garten zu, we’re talking about a specific group of children, so Die Kinder is used.

Is schauen here present simple or present continuous? Could it mean “are watching”?

German has only one present tense form (Präsens), and it corresponds to both:

English present simple:
- The children watch the dog.
English present continuous:
- The children are watching the dog.

So Die Kinder schauen dem Hund im Garten zu. can be translated naturally as either, depending on context. In most everyday contexts, English would probably use the continuous:
The children are watching the dog in the garden.

Is zu in zuschauen the same as the preposition zu (to, at)?

No, they are related historically but function differently here:

As a verb prefix (in zuschauen), zu- is part of the verb. It changes or specifies the meaning:
- schauen – to look
- zuschauen – to watch (actively observe)
As a prefix, it is separable in main clauses:
- Die Kinder schauen dem Hund zu.
As a preposition, zu means to / towards / at:
- Ich gehe zu dem Hund. – I go to the dog.
- Er kommt zu mir. – He comes to me.

In the sentence Die Kinder schauen dem Hund im Garten zu, zu is only the separable prefix of the verb zuschauen, not a preposition.