Learn Hardcore Turkish: Hafta içi kütüphanede ders çalışıyorum.

It means on weekdays (Monday–Friday). It refers to the workweek as opposed to the weekend. If you want to say sometime within this week (by the end of the week), use hafta içinde or bu hafta içinde, which is more like during/within this week and not specifically excluding the weekend.

Is Hafta içi written as one word, two words, or with a hyphen? How is it capitalized?

It’s written as two words: hafta içi (not haftaiçi).
No hyphen.
It’s not a proper noun, so normally lowercase: hafta içi. At sentence start, Hafta is capitalized as usual: Hafta içi.

Can I say hafta içleri or hafta içi her gün to emphasize habit?

Yes.

Hafta içleri = on weekdays (habitually). Common in speech for routines.
Hafta içi her gün = every weekday (more explicit). Similarly, for weekends you’ll hear hafta sonları (on weekends, habitually).

Why is the present continuous çalışıyorum used if this is a habitual routine?

Turkish often uses the present continuous (-yor) for current routines/schedules, especially when a time adverb like hafta içi is present. You could also use the aorist:

Hafta içi kütüphanede ders çalışıyorum. → I study on weekdays (this is my current routine).
Hafta içi kütüphanede ders çalışırım. → I (generally) study on weekdays (habit, rule-of-thumb). The aorist feels more generic/timeless; -yor feels like a present, ongoing schedule.

What does the set phrase ders çalışmak mean, exactly?

It’s the idiomatic way to say to study (for school/lessons). Literally lesson + work, but together it means to study. Compare:

ders çalışmak = to study (do one’s coursework/homework)
okumak = to read; also to study at (be a student at) a school/university
öğrenmek = to learn (acquire knowledge/skill)

Why isn’t ders marked with accusative (dersi)?

Because the object is indefinite and the phrase ders çalışmak behaves like a set expression. With a specific/definite lesson you add accusative:

Ders çalışıyorum. = I’m studying (in general).
Matematik dersini çalışıyorum. = I’m studying the math lesson (a particular one).

Can I drop ders and just say çalışıyorum?

Be careful: çalışmak by itself usually means to work (have a job). In some school contexts it can mean study, but to avoid ambiguity, use ders çalışmak when you mean study:

Kütüphanede çalışıyorum. → I work at the library.
Kütüphanede ders çalışıyorum. → I study at the library.

What is the suffix -de in kütüphanede?

It’s the locative case, meaning in/at: kütüphanede = in/at the library. It’s a suffix written attached to the noun. Don’t confuse it with the separate particle de/da meaning also/too, which is written separately (e.g., Ben de).

Why is it kütüphane-de (with -de) and not -da or -te/-ta?

Two rules:

Vowel harmony for -de/-da: last vowel of the noun is front (e), so choose -de.
Devoicing for -de/-da vs. -te/-ta: after a voiceless consonant, -de/-da become -te/-ta (e.g., parkta). Here the word ends in a vowel (kütüphane), so no devoicing: kütüphanede.

How would I say to the library and from the library?

Dative (to): kütüphaneye (note the buffer letter y because the noun ends in a vowel)
Ablative (from): kütüphaneden

Is the word order fixed? Could I move the time/place around?

Turkish is flexible. A common neutral order is Time–Place–Verb:

Hafta içi kütüphanede ders çalışıyorum. You can shuffle for emphasis:
Kütüphanede hafta içi ders çalışıyorum.
Hafta içi ders çalışıyorum kütüphanede. (puts extra emphasis on location) Meaning stays, but the element closest to the verb tends to be in focus.

Do I need to include the subject pronoun ben?

No. The verb ending -um already marks first person singular. Ben is added only for emphasis or contrast:

(Ben) hafta içi kütüphanede ders çalışıyorum.
Ben vurgusu: I (as opposed to others) study on weekdays at the library.

How do I turn this into a yes–no question like Do you study at the library on weekdays?

Use the question particle mi (separate word), after the verb:

2nd sg: Hafta içi kütüphanede ders çalışıyor musun?
2nd pl/formal: Hafta içi kütüphanede ders çalışıyor musunuz?

How do I negate it?

Insert the negative -me/-ma before -yor:

Hafta içi kütüphanede ders çalışmıyorum. Note vowel harmony: çalış-
- -m-
  - -ıyor-
    - -um → çalışmıyorum.

What about weekends? How would I say the weekend version?

Use hafta sonu (singular) or hafta sonları (habitual):

Hafta sonu kütüphanede ders çalışıyorum.
Hafta sonları kütüphanede ders çalışıyorum. (on weekends as a habit)

How is çalışıyorum built morphologically?

çalış- (work/study)
-(I)yor (present continuous; the initial vowel harmonizes: ı/i/u/ü → here -ıyor)
-um (1st person singular; its vowel harmonizes with the last vowel before it, the o in yor, so it’s -um, not -ım) Result: çalışıyorum.

Any quick pronunciation tips for the words here?

ç = ch (as in church)
ş = sh
ı (dotless i) = a central vowel (like the e in roses for many speakers)
ph in kütüphane is not like English ph=f; it’s p + h across a morpheme boundary, with an audible h.
Stress is typically on the last syllable with the locative: kütüphanede; çalışıyorum.

Can I say only on weekdays?

Yes: Sadece hafta içi kütüphanede ders çalışıyorum. You can also use yalnız instead of sadece.