Learn Hardcore German: Die Schülerin macht abends im Wohnzimmer ihre Hausaufgaben. - The pupil female does her homework in the living room in the evenings.

The subject of the sentence is die Schülerin (the girl student), so it must be in the nominative case.
Schülerin is grammatically feminine, so nominative singular feminine → die.

If the sentence were about a male pupil, it would be der Schüler (nominative masculine singular).

Why is the verb form macht used here?

The verb is machen (to do / to make).

The subject is die Schülerin = she (3rd person singular).
The 3rd person singular form of machen in the present tense is macht.

Conjugation (present tense) looks like this:

ich mache
du machst
er/sie/es macht
wir machen
ihr macht
sie machen

Why is it macht Hausaufgaben and not tut Hausaufgaben?

In German, some verbs just collocate (naturally go together) with certain nouns:

Hausaufgaben machen = to do homework (this is the normal, idiomatic expression)
Hausaufgaben tun sounds wrong or at least very unusual to native speakers.

So even though tun also means to do, with Hausaufgaben you almost always use machen.

Why is Hausaufgaben in the plural? Can you also say Hausaufgabe?

Yes, there is a singular:

die Hausaufgabe = one homework assignment
die Hausaufgaben = homework / several assignments

In everyday speech, Germans normally say die Hausaufgaben machen even if in English you would just say do homework, without making it plural.

So ihre Hausaufgaben here is like her homework in general, not necessarily counted individually.

What does ihre mean, and why is it ihre Hausaufgaben and not ihr Hausaufgaben?

ihre is a possessive determiner and here it means her.

Possessive determiners in German must agree with the gender, number and case of the noun they describe, not with the owner.

The owner is die Schülerin (she).
The thing owned is Hausaufgaben (plural, accusative, because it’s the direct object).

For ihr- (her), the accusative plural ending is -e, so:

ihre Hausaufgaben = her homework (plural, direct object)

If the noun were singular feminine in accusative, you’d also get ihre:

ihre Tasche (her bag – accusative or nominative feminine singular)

What case is ihre Hausaufgaben, and how can I see that?

ihre Hausaufgaben is in the accusative plural.

Reason: It is the direct object of the verb machen (What does she do? → her homework).

Clues:

The subject (die Schülerin) is nominative.
The direct object is usually accusative.
Hausaufgaben is plural; the accusative plural of die-nouns looks like the nominative plural (no extra change in the noun).

So the pattern is:

Nominative: die Hausaufgaben
Accusative: die Hausaufgaben

But the possessive shows that it’s plural accusative: ihre (acc. pl.) instead of ihr (e.g., dat. masc./neut. sg.).

Why is it im Wohnzimmer and not in dem Wohnzimmer?

im is simply the contracted form of in dem.

in (in) + dem (dative masculine/neuter singular) → im

Wohnzimmer is a neuter noun:

nominative: das Wohnzimmer
dative singular: dem Wohnzimmer

Because you use the dative after in when you are talking about location (where?), you get in dem Wohnzimmer, usually contracted to im Wohnzimmer.

So im Wohnzimmer = in the living room (at that place).

Why is Wohnzimmer capitalized, but abends is not?

In German:

All nouns are capitalized: Wohnzimmer, Schülerin, Hausaufgaben, etc.
Adverbs and most other parts of speech are not capitalized.

abends is an adverb meaning in the evenings / at night (regularly), so it is written with a lowercase initial letter.

Wohnzimmer is a noun (literally living room), so it must start with a capital letter.

What exactly does abends mean, and how is it different from am Abend?

abends is an adverb and usually expresses a habit or regularity:

Die Schülerin macht abends ihre Hausaufgaben.
→ She does her homework in the evenings / in the evening (as a regular habit).

am Abend literally means on the evening / in the evening and often refers to a specific evening/time frame, though context matters:

Heute Abend or am Abend → this evening / in the evening (today).

In many contexts, abends sounds more like a general routine; am Abend can sound more like one particular evening or the time of day more neutrally.

Why is the word order macht abends im Wohnzimmer ihre Hausaufgaben and not another order?

German main clauses have:

One finite verb in second position (Verb-second rule, V2).
Most other elements go into the Mittelfeld (middle field), where the order is fairly flexible but has common tendencies.

The core pattern here:

Die Schülerin (subject, 1st position)
macht (finite verb, 2nd position)
abends im Wohnzimmer ihre Hausaufgaben (time, place, object in the Mittelfeld)

A common guideline is Time – Manner – Place (TMP). Here:

abends → Time
im Wohnzimmer → Place
ihre Hausaufgaben → Object

So abends im Wohnzimmer feels natural because the time element comes early.
You could say:

Die Schülerin macht im Wohnzimmer abends ihre Hausaufgaben.

This is grammatically correct but a bit less neutral; it might sound like you are emphasizing the place more than the time.

Could I start the sentence with abends? How would that change things?

Yes, and it’s very common:

Abends macht die Schülerin im Wohnzimmer ihre Hausaufgaben.

Here:

Abends takes the first position.
The finite verb macht must still be in second position, so the subject die Schülerin moves after the verb.

This version puts more emphasis on the time frame (in the evenings) as the main point of the sentence.

Why is it die Schülerin and not eine Schülerin? Does die mean she’s known?

die Schülerin is the definite form: the (female) pupil.
eine Schülerin would be a (female) pupil.

Often, die Schülerin refers to a specific girl that speaker and listener both know about (for example, someone already mentioned in the conversation or visible in the situation).

However, German can also use die for a more general or typical person in stories or examples, similar to:

The student does her homework in the evening. (talking about a typical or example student)

Context decides whether she is specific or generic.

Is Hausaufgaben feminine, masculine, or neuter, and how do its forms work?

The base noun is die Hausaufgabe (feminine).

Singular:

nominative: die Hausaufgabe
accusative: die Hausaufgabe
dative: der Hausaufgabe
genitive: der Hausaufgabe

Plural:

nominative: die Hausaufgaben
accusative: die Hausaufgaben
dative: den Hausaufgaben
genitive: der Hausaufgaben

In the sentence you have ihre Hausaufgaben in the accusative plural.