Learn Hardcore German: Ich lese ein Kapitel pro Tag.

Because Kapitel is a neuter noun (das Kapitel). In the accusative case, the neuter indefinite article stays ein. Only masculine nouns take einen in the accusative (e.g., einen Roman). Feminine would be eine (e.g., eine Seite).

What case is Kapitel in here?

Accusative. Lesen takes a direct object in the accusative: Ich lese ein Kapitel. Compare:

Masculine: Ich lese einen Artikel.
Feminine: Ich lese eine Nachricht.
Neuter: Ich lese ein Kapitel.

What exactly does pro Tag do, and do I need an article?

Pro means “per” and forms a rate expression with a bare singular noun: pro Tag, pro Woche, pro Person. You don’t use an article here: not pro dem Tag. It’s a set measurement phrase, like English “per day.”

Can I say jeden Tag or täglich instead of pro Tag?

Yes:

Ich lese jeden Tag ein Kapitel. (every day; very common in speech)
Ich lese täglich ein Kapitel. (daily; a bit more compact/formal)
Ich lese ein Kapitel pro Tag. (per day; often used with quantities, rates, and in writing/ads) All are correct; the nuance is tiny. Pro Tag highlights the rate; jeden Tag/täglich highlights the habitual frequency.

Is Ich lese ein Kapitel jeden Tag okay, or is the time phrase supposed to come earlier?

It’s okay. By the common ordering guideline (often taught as Te-Ka-Mo-Lo: time–cause–manner–place), a time expression like jeden Tag usually comes earlier: Ich lese jeden Tag ein Kapitel. But Ich lese ein Kapitel jeden Tag is acceptable and may put a bit more focus on the object first.

Can I start with the time phrase, like Pro Tag lese ich ein Kapitel?

Yes. German main clauses are verb-second (V2), so if you front a time phrase, the finite verb stays in second position:

Pro Tag lese ich ein Kapitel.
Jeden Tag lese ich ein Kapitel.
All of these are natural word orders.

Why are Kapitel and Tag capitalized?

All nouns are capitalized in German: das Kapitel, der Tag. The pronoun ich is normally lowercase, but it’s capitalized at the start of a sentence (as here). The formal Sie is always capitalized.

How do I pronounce the sentence naturally?

Ich: the soft “ich-sound” [ç], not like English “sh.”
lese: two clear vowels: LEH-zeh.
Kapitel: stress the second syllable: ka-PI-tel.
pro: long o, like “proh.”
Tag: long a [aː]; the final g is pronounced like a hard k: tahk.
Altogether: roughly “Ish LAY-zeh eye-n ka-PEE-tel pro tahk.”

What’s the plural of Kapitel?

It’s the same as the singular: die Kapitel. Examples:

Singular: ein Kapitel
Plural: zwei Kapitel You’d say: Ich lese zwei Kapitel pro Tag.

How do I negate this naturally?

Use kein to negate the indefinite noun phrase:

Ich lese kein Kapitel pro Tag. (I don’t read a chapter per day.) If you want to negate the time-frequency idea instead, you can say:
Ich lese nicht jeden Tag ein Kapitel. (I don’t read a chapter every day.)

Can I drop ich and just say Lese ein Kapitel pro Tag?

No. German normally requires the subject pronoun. You can drop it only in imperatives or in very informal ellipses (e.g., note-taking), not in standard statements.

How would I say this as a future plan?

Use werden + infinitive:

Ich werde jeden Tag ein Kapitel lesen.
Ich werde pro Tag ein Kapitel lesen. In many contexts, present tense already works for planned/habitual actions, so the original sentence can also imply an ongoing routine.

Is per Tag or je Tag also correct?

per Tag is understood and used, especially in pricing (e.g., 50 Euro per Tag).
pro Tag is more natively German in everyday phrasing.
je is common in distributive contexts with a number or plural: 1 Ticket je Person, zwei Aufgaben je Woche. Ein Kapitel je Tag is possible but sounds more formal/technical than pro Tag.

What’s the difference between pro Tag, am Tag, and tagsüber?

pro Tag = per day (a rate).
am Tag = in/within the day, or used with counts: zweimal am Tag (twice a day). Saying ein Kapitel am Tag can be understood, but pro Tag or jeden Tag is clearer for a rate/habit.
tagsüber = during the daytime (as opposed to at night): Ich lese tagsüber. (I read during the day.)