Learn Hardcore Dutch: Anna studeert fotografie, dus zij neemt altijd haar camera mee. - Anna studies photography, so she always takes her camera with her.

studeren = to study formally or academically (e.g. pursuing a course or subject at school/university)
leren = to learn, to acquire knowledge or a skill in general
In English you often say “is studying,” but Dutch uses the simple present. So Anna studeert fotografie means “Anna studies (is studying) photography” in an academic/program context.

Why is fotografie used without an article, and why is there no aan before it?

When naming a field of study or program, Dutch drops the article: studeren fotografie, studeer biologie.
Use aan only when specifying the institution: studeer aan de universiteit (“study at the university”). You study a subject directly, but you study at a place.

What’s the function of dus, why is there a comma before it, and why doesn’t it cause inversion (it’s dus zij neemt, not dus neemt zij)?

dus = “so” or “therefore,” a coordinating conjunction/adverb linking two main clauses
The comma is optional but common to show a natural pause
Unlike subordinating conjunctions (e.g. omdat, dat), coordinating conjunctions like dus, en, maar do not send the verb to the end. Normal word order (subject–verb) stays in place.

Why is the verb meenemen split into neemt … mee in this sentence?

meenemen is a separable verb with prefix mee-
In main clauses the prefix detaches and moves to the end:
Anna neemt haar camera mee.
In subordinate clauses the prefix stays attached (e.g. “om haar camera mee te nemen”).

Why is altijd placed before haar camera mee instead of somewhere else?

In Dutch main clauses, adverbs of frequency (like altijd) typically follow the finite verb.
The default order is: Subject – finite verb – adverb – object/rest.
Anna (S) – neemt (V) – altijd (Adv) – haar camera mee (O).

Why do studeert and neemt both end in -t, and what are the conjugation rules here?

In the present tense, the third person singular (hij/zij/het) adds -t to the stem:
Ik studeer / jij studeert / hij studeert
Ik neem / jij neemt / hij neemt
Note that nemen is irregular in the first person (ik neem) but follows the regular pattern (stem + -t) in 2nd/3rd person.

Why is zij used here instead of the shorter ze, and is it optional to drop the pronoun altogether?

zij is the full subject pronoun; ze is its reduced, more colloquial form.
Both are correct: “dus zij neemt” or “dus ze neemt.”
You cannot drop the subject pronoun entirely in a Dutch main clause; it needs an explicit subject.

Why is it haar camera, and how do Dutch possessive pronouns work?

haar is the invariable possessive pronoun for “her.” It agrees with the possessor (Anna), not the camera.
Dutch possessives reflect the person/number of the owner, never get an apostrophe: always haar camera.