Learn Hardcore German: Sie rettete mutig den Hund aus dem Wasser. - She bravely rescued the dog from the water.

Because it’s the direct object (accusative). Masculine definite article changes: nominative der Hund → accusative den Hund. Indefinite: ein Hund → einen Hund.

Why dem Wasser after aus?

Because aus always governs the dative. Neuter das Wasser becomes dem Wasser. Memory tip: dative prepositions include aus, bei, mit, nach, seit, von, zu (and gegenüber).

What’s the difference between aus dem Wasser and vom Wasser?

aus dem Wasser = out of the water (from inside to outside). Fits a rescue.
vom Wasser = from the water (as a source/cause), e.g., vom Wasser durchnässt (wet from the water). Not used for pulling someone out.

Is the word order with mutig natural?

Yes, but many speakers would place the object earlier: Sie rettete den Hund mutig aus dem Wasser. The given order (Sie rettete mutig den Hund …) is possible and can foreground the manner.

Where else can mutig go, and what changes?

Mutig rettete sie den Hund aus dem Wasser. Emphasis on bravery (fronted).
Sie rettete den Hund mutig aus dem Wasser. Neutral/common.
Sie rettete den Hund aus dem Wasser mutig. Grammatically possible but sounds awkward; manner adverbs rarely sit at the very end. Rule of thumb inside the clause: Te-Ka-Mo-Lo (time–cause–manner–place). Here, mutig (manner) before aus dem Wasser (place) fits.

Why use simple past (rettete) instead of present perfect?

In everyday speech many prefer the present perfect: Sie hat den Hund mutig aus dem Wasser gerettet. Simple past (rettete) is common in writing and in northern German; both are correct.

How is rettete formed and pronounced?

Verb: retten (regular/weak).
Preterite adds -te and an extra -e- because the stem ends in -t: ich rettete (not ×ich rettte).
Pronunciation: three syllables, roughly “RET-te-te.”

Is mutig an adjective or an adverb here? Why no “-ly” ending?

It’s an adjective used adverbially. German typically uses the plain adjective as an adverb: schnell fahren, leise sprechen, mutig retten. Mutigerweise also exists but is more formal and less common.

Could I say die Hündin instead of den Hund?

Yes, if you specifically mean a female dog: accusative feminine is die Hündin (unchanged from nominative). Hund is masculine and can be generic; Hündin specifies female.

Do I need the articles? Why not Sie rettete Hund aus Wasser?

Standard German normally requires articles with countable nouns: den Hund. Without the article, Hund sounds telegraphic. With Wasser, aus Wasser means “made of water,” not “out of the water.” For location, keep aus dem Wasser.

If I replace the object with a pronoun, where does it go?

Pronoun objects tend to come early:

Sie rettete ihn mutig aus dem Wasser.
In the perfect: Sie hat ihn aus dem Wasser gerettet.

Are there common variants with heraus?

You can add a directional particle for emphasis:

Sie rettete den Hund aus dem Wasser heraus. (a bit redundant but idiomatic) More common with other verbs: Sie zog den Hund aus dem Wasser heraus. Herausretten is rare; prefer retten or herausholen/ziehen.

What’s the typical order of adverbials, and how does this sentence fit?

Te-Ka-Mo-Lo (time–cause–manner–place). Here:

Manner: mutig
Place: aus dem Wasser So … mutig … aus dem Wasser fits. With time: Sie rettete gestern mutig den Hund aus dem Wasser.

How would I negate this sentence?

Place nicht before what you negate:

General/place: Sie rettete den Hund nicht aus dem Wasser (sondern vom Eis).
Object contrast: Sie rettete nicht den Hund aus dem Wasser, sondern die Katze.

Is the capitalization correct?

Yes. All nouns are capitalized: Hund, Wasser. Verbs and adjectives are lowercase: rettete, mutig.