Learn Hardcore Korean: geunyeoneun jumalmada doseogwaneseo hangugeoreul gongbuhaeyo. - She studies Korean at the library every weekend.

는 marks the topic: it frames what you’re talking about (as for her). 그녀가 uses the subject marker 가, which highlights who performs the action, often in answer to who-questions or with contrastive emphasis. Both are grammatical:

그녀는 sets the stage to talk about her routine.
그녀가 would stress that she (as opposed to someone else) is the one who studies.

Is 그녀 natural in everyday Korean?

Not really. 그녀 is common in textbooks and some writing, but in conversation people usually:

Use a name or title.
Say 그 사람 (that person), 그 여자 (that woman), or polite 그분.
Omit the subject if clear from context. Casual speech may use 걔 (she/he, for someone familiar and younger/same age).

What does -마다 in 주말마다 mean? Can I use 주말에 or 매주 instead?

-마다 means every/each, and it attaches directly to a noun: 주말마다 = every weekend.

주말에 = on the weekend (not necessarily every weekend).
매주 = every week; to mean every weekend you can say 매주 주말에. All are natural; choose based on whether you want every weekend (주말마다) or just on weekends (주말에).

Why is it 도서관에서 and not 도서관에?

에서 marks the place where an action happens (at/in while doing). 에 marks location/existence or destination/time.

도서관에서 공부해요 = studies at the library.
도서관에 가요 = goes to the library.

What’s the typical word order, and can I move things around?

A common order is Topic/Subject + Time + Place + Object + Verb. Korean allows flexibility as long as the verb stays at the end:

주말마다 도서관에서 한국어를 공부해요 (very natural)
한국어를 도서관에서 공부해요 (object emphasized)
도서관에서 한국어를 주말마다 공부해요 (less typical but acceptable) Moving parts can shift emphasis; keep the verb final.

What does 를 in 한국어를 do, and why 를 instead of 을?

을/를 marks the direct object. Use 를 after a vowel and 을 after a consonant. 한국어 ends in a vowel sound, so 한국어를 is correct.

Can I drop the object particle 를 here?

Yes, in casual speech it’s common: 한국어 공부해요. Omitting 를 is natural when the meaning is clear. Keeping it is safer for learners and sounds a bit more careful/explicit.

Should I use 공부해요 or 배워요 for languages?

Both are possible but differ slightly:

공부하다 = study (often self-directed or general studying).
배우다 = learn (implies being taught or taking a class). At a library context, 공부해요 fits well. If you mean she is taking Korean classes, 한국어를 배워요 is great.

What politeness level is 공부해요? What are alternatives?

공부해요 is the standard polite -아요/어요 style.

Formal polite: 공부합니다 (presentational, announcements, news).
Casual/informal: 공부해 (friends, close peers). Choose based on who you’re talking to.

Does this sentence mean she is studying right now or that she studies habitually?

Here it’s habitual because of 주말마다. The plain present -아요/어요 covers both simple present and present progressive depending on context. To say she’s doing it right now: 공부하고 있어요.

Can I omit the subject entirely?

Yes. Korean drops known subjects frequently. 주말마다 도서관에서 한국어를 공부해요 is perfectly natural if it’s clear who you’re talking about.

Are 한국어 and 한국말 the same?

Largely yes.

한국어 is a bit more formal/standard (used in course names, writing).
한국말 sounds more colloquial/everyday. Both work in this sentence.

How do I say only on weekends instead of every weekend?

Use -만: 주말에만 도서관에서 한국어를 공부해요 = she studies only on weekends (not on weekdays). Note that 주말에만 doesn’t necessarily mean every weekend; it just restricts the time to weekends. If you want both ideas, you can say the restriction explicitly: 주말마다 공부해요. 평일에는 안 해요.

Any pronunciation tips for this sentence?

Yes, some linking/assimilation occurs:

도서관에서 is pronounced with the ㄴ linking forward: roughly (도서과네서).
한국어를 assimilates to roughly (한구거를). These are natural sound changes; just say the words smoothly together.

How do I change it to past, future, or negative?

Past: 그녀는 주말마다 도서관에서 한국어를 공부했어요.
Future/intention: 그녀는 주말마다 도서관에서 한국어를 공부할 거예요.
Negative: 그녀는 주말마다 도서관에서 한국어를 공부하지 않아요 or more colloquial 공부 안 해요.