Meine Nichte malt das Plakat mit einem roten Buntstift aus, und danach steckt sie den Spitzer wieder ins Federmäppchen.

Nichte is a feminine noun, so the possessive mein has to agree with it.

Here, meine Nichte is the subject of the sentence, so it is in the nominative case.
For a feminine singular noun in the nominative, mein- becomes meine.

• mein Neffe = my nephew
• meine Nichte = my niece

Because the verb is ausmalen, which is a separable verb.

In a normal main clause, the conjugated part goes in the usual verb position, and the prefix moves to the end:

• Sie malt das Plakat aus.

So:

• malt = the conjugated verb
• aus = the separable prefix

This verb means something like to color in or to fill in with color, not just to paint/draw in a general sense.

Malen is the general verb to paint or to draw.
Ausmalen usually means to color in something, especially an area or shape.

So:

• ein Bild malen = to paint/draw a picture
• ein Bild ausmalen = to color in a picture

In this sentence, ausmalen suggests that she is coloring the poster in with the pencil, not just making a drawing on it.

Das Plakat is the direct object, so it is in the accusative case.
However, Plakat is a neuter noun, and for neuter singular nouns, the definite article is:

• das in the nominative
• das in the accusative

So the form does not change.

Compare:

• Das Plakat ist groß. = nominative
• Sie malt das Plakat aus. = accusative

Because the preposition mit always takes the dative case.

So Buntstift has to be in the dative:

• nominative: ein roter Buntstift
• dative: mit einem roten Buntstift

What changes here:

• ein → einem
• rot → roten

That is because Buntstift is masculine.

This is an adjective ending question.

In mit einem roten Buntstift:

• mit requires the dative
• Buntstift is masculine singular
• after einem, the adjective takes the ending -en

So:

• ein roter Buntstift = nominative
• mit einem roten Buntstift = dative

This is a very common pattern in German adjective declension.

German often combines words into a compound noun.

Buntstift is made from:

• bunt = colorful / colored
• Stift = pencil, pen, stick-like writing tool

Together, Buntstift means colored pencil.

In German, compounds are normally written as one word, not as separate words.

Because German main clauses follow the verb-second rule.

That means the conjugated verb must come in the second position, even if something other than the subject comes first.

So:

• Sie steckt danach den Spitzer ins Federmäppchen.
• Danach steckt sie den Spitzer ins Federmäppchen.

Both are possible, but if danach comes first, then steckt must come second, and the subject sie comes after it.

Because Spitzer is the direct object of steckt, so it is in the accusative case.

Spitzer is a masculine noun, and masculine nouns change like this:

• nominative: der Spitzer
• accusative: den Spitzer

So:

• Der Spitzer ist neu. = subject
• Sie steckt den Spitzer ins Federmäppchen. = direct object

Ins is a contraction of in das.

Here, German uses in + accusative because there is movement toward the inside of something:

• ins Federmäppchen = into the pencil case

By contrast, im is in dem, and it is used for location:

• Der Spitzer ist im Federmäppchen. = The sharpener is in the pencil case.

So the contrast is:

• ins = into it
• im = in it

Wieder can mean again, but in many contexts it also has the sense of back.

In this sentence, wieder means that she puts the sharpener back into the pencil case, probably where it normally belongs.

So the idea is not just repetition, but return to the earlier place/state.

Stecken is often used when something is put into a container, slot, pocket, case, or other enclosed space.

So den Spitzer ins Federmäppchen stecken sounds very natural:
she is putting/sticking the sharpener into the pencil case.

Legen means to lay/place, and it can also be possible in some contexts, but stecken fits especially well when the object goes into something like a bag, pocket, or case.

Usually, no.

When und connects two main clauses, German normally does not require a comma:

• Meine Nichte malt das Plakat mit einem roten Buntstift aus und danach steckt sie den Spitzer wieder ins Federmäppchen.

A comma can sometimes appear for readability or style, but in standard neutral writing, many people would simply leave it out here.