Learn Hardcore Korean: doseogwan chaegeun jikjeop jettae bannaphaeya hae. - You must return library books yourself on time.

은/는 marks the topic: it sets up “as for library books…” and often sounds like a general rule or contrastive statement.
Using 을/를 would mark it simply as the object with no topical nuance: 도서관 책을 직접 제때 반납해야 해.
Both are grammatical. 은 feels like you’re stating a policy/guideline (“When it comes to library books…”), while 을 is a straightforward statement about what must be done.

Who is the implied subject? There’s no “I/you/we” in the sentence.

Korean often drops pronouns. The subject is recovered from context:

In conversation, 해야 해 commonly targets the listener, so it often means “you must…”
It can also mean “I/we must…” if that’s clear from context.
On signs/notices, you’d usually see the more formal 해야 합니다 addressing the general public.

What exactly does 직접 mean? Is it the same as 스스로 or 혼자서?

직접: in person, without a proxy or intermediary. Emphasizes doing it yourself rather than having someone/something else do it.
스스로: by oneself, of one’s own accord/effort (self-driven). Not about intermediaries, but about self-reliance.
혼자서: alone, without company or help (physically by yourself). Examples:
우편이나 대리인 말고 직접 반납해. (Not by mail or another person.)
숙제는 스스로 해야 해. (Self-driven effort.)
혼자서 도서관에 갔다. (Went alone.)

What does 제때 mean exactly? Is 제때에 also correct? Any synonyms?

제때 means “at the right time; on time.” It already functions adverbially.
제때에 is also correct (a bit more formal/explicit).
Synonyms: 제시간에, 기한 내에, 정해진 날짜까지, 시간 맞춰서.
Spelling tip: it’s one word, 제때 (not “제 때”).

Is the order 직접 제때 natural? Could I say 제때 직접 instead?

Both are acceptable. Time adverbs often come before manner adverbs, so 제때 직접 can feel slightly more neutral. 직접 제때 puts a bit more early emphasis on the “in person” part. Meaning doesn’t change.

Why use 반납하다 instead of 돌려주다?

반납하다: to return something you borrowed to an institution (library, office, school). This is the standard verb for library materials.
돌려주다: to give something back to a specific person.
A more formal synonym is 반환하다 (common in official notices).

How does 해야 해 express obligation? What’s the difference from 해야 돼 or 해야 한다?

Pattern: A–아/어야 하다 = “must/should.” Here, 반납해야 하다 → colloquial 반납해야 해.
해야 돼(요) (from 되다) is also very common; many speakers use 해야 해(요) and 해야 돼(요) interchangeably.
해야 한다 is more formal and can sound firmer/stronger; 해야 합니다 is the polite formal version.

What politeness level is this? How would I say it more politely or more formally?

Current sentence: casual 해 style.
Polite: 도서관 책은 직접 제때 반납해야 해요.
Formal polite (announcements, rules): 도서관 책은 직접 제때 반납해야 합니다.
If you want honorifics toward the subject: 반납하셔야 해요/합니다.

Why is there no plural marker 들 on 책? Should it be 책들?

Korean usually omits plural 들 unless you need to stress multiplicity. 책 can mean “book(s)” from context.
책들 is possible but can sound marked; here, a general rule doesn’t need 들. 책은 naturally covers all relevant books.

Why is it 도서관 책 and not 도서관의 책? Are both okay?

Noun–Noun compounds are very common in Korean: 도서관 책 (“library books”) is natural and common in speech.
도서관의 책 is not wrong but can feel more formal or stiff in everyday use.
If you want to be explicit: 도서관에서 빌린 책 (“books borrowed from the library”).

Could I say 도서관은 책을 직접 제때 반납해야 해? What changes?

That topicalizes the library itself: “As for the library, (it) must return the books,” which is a different (and odd) meaning here. You want the books as the topic/object, not the library.

Is the object particle missing? Should it be 책을?

Because 은 is marking topic on 책, you don’t also use 을. If you drop topicalization, you can say:

도서관 책을 직접 제때 반납해야 해. Both versions are correct; they differ only in the topical nuance.

Any pronunciation tips?

도서관 책은 → [도서관 채근] (the ㅋ in 책 links to the vowel of 은: 책은 → 채근)
직접 → [직쩝] (tensification of ㅈ)
제때 → [제때]
반납해야 해 → commonly [반나패야 해] (ㅎ-triggered aspiration across the boundary: 반납해-) Speak it smoothly as one rhythm group.

How can I add emphasis like “definitely” or “make sure to”?

꼭 (colloquial): 도서관 책은 꼭 직접 제때 반납해야 해요.
반드시 (more formal/strong): 도서관 책은 반드시 직접 제때 반납해야 합니다.
You can place 꼭/반드시 before the adverbs or right before the verb phrase.

If I want to be explicit that the books are borrowed, how do I say that?

도서관에서 빌린 책은 직접 제때 반납해야 해.
More formal/institutional: 도서관에서 대출한 책은 직접 제때 반납해야 합니다.