Learn Hardcore German: Meine Schwester malt Kaninchen, Hamster und Fische aus unserem Aquarium in ihr Skizzenbuch. - My sister paints rabbits, hamsters and fish from our aquarium in her sketchbook.

Schwester is a feminine noun (die Schwester).
As the subject, it stands in the nominative case.
The form meine (not meiner / meiner Schwester etc.) tells you it’s nominative feminine singular.

So Meine Schwester = my sister and is the nominative subject that performs the action malt (paints).

Why is the verb malen written as malt here?

Malt is just the conjugated form of the infinitive malen (to paint).

Malen is a regular verb:

ich male
du malst
er / sie / es malt
wir malen
ihr malt
sie / Sie malen

The subject is meine Schwester (3rd person singular), so the correct present‑tense form is sie malt = she paints.

Why are Kaninchen, Hamster und Fische written without any articles like die or einige?

German often drops the article with indefinite plural nouns when you mean “some” or talk in general:

Meine Schwester malt Kaninchen, Hamster und Fische.
→ My sister paints rabbits, hamsters and fish. (some, in general)

You could add articles, but the meaning changes slightly:

… malt die Kaninchen, die Hamster und die Fische …
→ specific, known animals: the rabbits, the hamsters and the fish…
… malt einige Kaninchen …
→ paints a few rabbits…

So the sentence without articles suggests a non‑specific, general group.

How can I tell whether Kaninchen is singular or plural here?

The noun Kaninchen has the same form in singular and plural:

Singular: das Kaninchen – the rabbit
Plural: die Kaninchen – the rabbits

In this sentence, context tells you it’s plural:

It’s part of a list: Kaninchen, Hamster und Fische.
In English you naturally say rabbits, hamsters and fish, not a rabbit, a hamster and a fish in this generic description.

So here Kaninchen is understood as rabbits.

What case are Kaninchen, Hamster und Fische, and how can I see that?

They are direct objects of the verb malt, so they are in the accusative case.

In this particular sentence, the forms look exactly like the nominative:

Kaninchen (same in nominative and accusative)
Hamster (same in nominative and accusative)
Fische (plural: nominative and accusative are both Fische)

You would see the case more clearly if there were articles:

Nominative: Die Kaninchen mögen Wasser.
Accusative: Meine Schwester malt die Kaninchen.

Here, their function (what is being painted?) tells you they are accusative objects.

What does aus unserem Aquarium do grammatically, and which case is used?

Aus unserem Aquarium is a prepositional phrase indicating the origin/source of the animals: they come from our aquarium.

aus is a preposition that always takes the dative case.
Aquarium is neuter: das Aquarium.
Dative singular neuter is dem Aquarium.

With the possessive unser‑, the forms are:

Nominative neuter: unser Aquarium
Accusative neuter: unser Aquarium
Dative neuter: unserem Aquarium

Because aus requires the dative, we get aus unserem Aquarium = from our aquarium.

Why is it unserem Aquarium and not unser Aquarium or unseren Aquarium?

The form of unser‑ depends on:

Case – here: dative (because of aus)
Gender and number of the noun – here: neuter singular (das Aquarium)

Dative singular forms of unser‑:

Masculine: unserem Vater
Neuter: unserem Kind
Feminine: unserer Mutter

Since Aquarium is neuter, dative singular becomes unserem Aquarium.

unser Aquarium would be nominative/accusative, not dative.
unseren Aquarium would be wrong; -en is not the dative ending for neuter with this type of determiner.

Could I say von unserem Aquarium instead of aus unserem Aquarium?

You can say von unserem Aquarium, but the nuance changes slightly:

aus unserem Aquarium focuses on from the inside of our aquarium (the animals live in it).
von unserem Aquarium is more like from our aquarium in a looser sense: from that place / belonging to that aquarium, not so clearly “out of the inside”.

In this context, since fish are physically inside the aquarium, aus unserem Aquarium is the most natural and precise choice.

Is aus here part of a separable verb like ausmalen, or is it just a preposition?

Here aus is a preposition, not part of a separable verb.

The verb in the sentence is malt (from malen).
The phrase aus unserem Aquarium clearly modifies the animals: Kaninchen, Hamster und Fische aus unserem Aquarium = rabbits, hamsters and fish from our aquarium.

If it were the separable verb ausmalen (to color in), you would normally see a structure like:

Meine Schwester malt die Fische aus.
(My sister colors in the fish.)

In that case, aus would stand at the end of the clause, and there would not be a noun like Aquarium immediately after aus. So in your sentence, aus belongs with Aquarium, not with malt.

Why is it in ihr Skizzenbuch and not in ihrem Skizzenbuch?

The preposition in can take accusative or dative:

Accusative = movement into / direction (into)
Dative = location / position (in, inside)

Here, in ihr Skizzenbuch is understood as direction: she paints the animals into her sketchbook (onto its pages), so German uses the accusative:

Feminine accusative with ihr‑: ihr Skizzenbuch

If you said in ihrem Skizzenbuch (dative), it would emphasize the place where something is already located: in her sketchbook. That is possible with some verbs, but for the idea “she paints them into her sketchbook”, in ihr Skizzenbuch with accusative is the natural form.

What exactly does ihr mean here, and how do I know it means her and not their or your?

The possessive ihr can mean:

her (3rd person singular feminine)
their (3rd person plural)
your (formal, addressing one or more people: Ihr)

In the sentence:

The subject is meine Schwester (my sister), singular and feminine.
The most natural reading is that the sketchbook belongs to her: my sister paints … in her sketchbook.

There is no plural group mentioned that ihr could refer to as their, and there is no addressee that would make sense for formal Ihr.

So here ihr Skizzenbuch clearly means her sketchbook.

Is the word order aus unserem Aquarium in ihr Skizzenbuch fixed, or can I change it?

German allows quite a bit of flexibility in the middle field (everything between the finite verb and the clause-final elements).

Your sentence has:

Direct objects: Kaninchen, Hamster und Fische
Source PP: aus unserem Aquarium
Goal PP: in ihr Skizzenbuch

The given order (source → goal) is very natural: we often mention where something comes from before where it goes to.

Other orders are possible, for example:

Meine Schwester malt in ihr Skizzenbuch Kaninchen, Hamster und Fische aus unserem Aquarium.
Meine Schwester malt Kaninchen, Hamster und Fische in ihr Skizzenbuch aus unserem Aquarium. (sounds more marked / less elegant)

They are grammatically possible, but native speakers usually prefer:

Verb in second position (malt)
Objects and prepositional phrases in an order that feels logical (here: animals → from where → into what)

So aus unserem Aquarium in ihr Skizzenbuch is not absolutely fixed, but it is a very natural ordering.

Why is Skizzenbuch written as one word and capitalized?

German often creates compound nouns by putting several nouns together:

die Skizze (sketch)
das Buch (book)
→ das Skizzenbuch (sketchbook)

Rules:

Nouns in German are always capitalized, including compounds.
In compounds, the last part determines gender and plural:
- das Skizzenbuch (neuter, because Buch is neuter)
- plural: die Skizzenbücher

So Skizzenbuch is one compound noun, written as a single capitalized word.

Why does Fisch become Fische, but Kaninchen and Hamster do not change in the plural?

German has several common plural patterns; different nouns follow different ones.

Fisch → Fische
- Singular: der Fisch
- Plural: die Fische (add ‑e)
Hamster → Hamster
- Singular: der Hamster
- Plural: die Hamster (no change in the noun; only the article shows plural)
Kaninchen → Kaninchen
- Singular: das Kaninchen
- Plural: die Kaninchen (no change in the noun)

So:

Fische shows its plural with an ‑e ending.
Kaninchen and Hamster show their plural mainly through the article (die Kaninchen, die Hamster) and context, not by changing the noun form itself.