Learn Hardcore German: Jeden Abend schreibt sie drei Momente der Dankbarkeit in ihr Tagebuch. - Every evening she writes three moments of gratitude in her diary.

In this sentence, Jeden Abend is in the accusative case because German often uses the accusative for adverbial time expressions (how often? how long? when in a recurring sense?). So:

Nominative (subject): jeder Abend – every evening (as subject)
Accusative (here, time expression): jeden Abend – every evening

So you say Jeden Abend schreibt sie … = Every evening she writes …

Could I also say Sie schreibt jeden Abend drei Momente der Dankbarkeit in ihr Tagebuch? Is the meaning different?

Yes, that word order is completely correct:

Sie schreibt jeden Abend drei Momente der Dankbarkeit in ihr Tagebuch.

German main clauses are “verb‑second” clauses: the finite verb must be in the second position, but almost anything can be in the first position. So you have two natural options:

Jeden Abend schreibt sie drei Momente … (time expression in first position, more emphasis on when)
Sie schreibt jeden Abend drei Momente … (subject in first position, more neutral)

The basic meaning is the same; the first version slightly highlights the routine “every evening”.

Why does the verb (schreibt) come before the subject (sie) in Jeden Abend schreibt sie …?

In German main clauses, the finite verb is always in second position (the “V2 rule”). The “first position” is usually one single element (subject, time phrase, object, etc.).

Here, the first position is taken by Jeden Abend. That forces the verb into the second position, so schreibt must come next. The subject sie then comes after the verb:

Position: Jeden Abend
Position: schreibt (finite verb)
Rest: sie drei Momente der Dankbarkeit in ihr Tagebuch

If the subject is first, you’d get Sie schreibt jeden Abend …, but the verb is still second.

Why is there no article before drei Momente? Why not die drei Momente?

In German, when you use a cardinal number (eins, zwei, drei, …) directly before a noun, you usually don’t add a definite article unless you want to be very specific or contrast something.

drei Momente = three moments (in general, just saying how many)
die drei Momente = the three moments (those specific three, assumed to be known from context)

Here we’re just describing a routine: she writes three moments every evening, not referring to three particular, already-known moments. So drei Momente is natural.

What case is der Dankbarkeit in drei Momente der Dankbarkeit, and what does it do?

Der Dankbarkeit is genitive singular of die Dankbarkeit (“gratitude”).

Nominative: die Dankbarkeit
Genitive: der Dankbarkeit

In drei Momente der Dankbarkeit, the genitive phrase der Dankbarkeit describes what kind of moments they are: moments of gratitude. This is a common pattern:

ein Gefühl der Angst – a feeling of fear
ein Tag der Freude – a day of joy
drei Momente der Dankbarkeit – three moments of gratitude

Could I say drei dankbare Momente instead of drei Momente der Dankbarkeit? Is there a difference?

You can say drei dankbare Momente, but the nuance is slightly different:

drei Momente der Dankbarkeit literally: three moments of gratitude. The focus is on the abstract feeling “gratitude”.
drei dankbare Momente literally: three grateful moments. Grammatically, dankbare is an adjective describing the moments, which can sound a bit more like “grateful times”.

In everyday usage, drei Momente der Dankbarkeit sounds a bit more natural and idiomatic for the kind of journaling practice you mean. drei Dinge, für die sie dankbar ist (“three things she is grateful for”) is another very common phrasing.

Why is it in ihr Tagebuch and not in ihrem Tagebuch?

The preposition in is a so‑called “two‑way preposition” (Wechselpräposition). It can take either:

Dative – location (where something is)
Accusative – direction/movement (into/onto something)

Here, she writes something into her diary, i.e. there is a movement towards the diary as a container. That requires accusative:

in ihr Tagebuch – into her diary (accusative, direction)

If you were describing a location (where something is already written), you’d use the dative:

In ihrem Tagebuch stehen viele Einträge. – There are many entries in her diary. (dative, location)

How can I generally tell whether in takes accusative or dative?

A common rule:

Dative = Wo? (Where?) – no movement, just location
Accusative = Wohin? (Where to?) – movement into/onto a place

Examples:

in der Tasche (dative): Wo? – in the bag (just sitting there)
in die Tasche (accusative): Wohin? – into the bag (moving something there)

In your sentence:

Wohin schreibt sie die Momente? – in ihr Tagebuch (accusative)

That’s why you get ihr (accusative neuter) instead of ihrem (dative neuter).

Could I say auf ihr Tagebuch instead of in ihr Tagebuch?

No, not with the same meaning.

in ihr Tagebuch schreiben = to write in/into her diary (on the pages inside the book)
auf ihr Tagebuch schreiben would literally mean “write on top of her diary” (physically on the cover), which is not what you intend.

For writing inside a book, notebook, or diary, German uses in.

Why is it ihr Tagebuch and not das Tagebuch?

ihr is a possessive pronoun meaning her (or their, or your‑formal, depending on context). Here, from the English meaning, we know it’s “her diary”.

das Tagebuch = the diary (no information about whose diary)
ihr Tagebuch = her diary

Possessive pronouns in German function like articles and must match the gender, number, and case of the noun:

Nominative neuter: ihr Tagebuch
Accusative neuter: also ihr Tagebuch (same form)

Since Tagebuch is neuter and used here in the accusative with in (direction), ihr is the correct form.

Could sie here mean “they” or “you (formal)”? How do I know it is “she”?

Formally, sie (lowercase s) can mean:

she (3rd person singular)
they (3rd person plural)

And Sie (capital S) can mean:

you (formal, singular or plural)

In your sentence:

It’s written with a lowercase sie, so it’s not the formal Sie.
The verb is schreibt, the 3rd person singular form of schreiben. For they, it would be sie schreiben.

So sie schreibt can only mean she writes.

Why are words like Abend, Momente, Dankbarkeit, and Tagebuch capitalized?

In German, all nouns are capitalized, no matter where they appear in the sentence. That includes:

Concrete things: der Abend, das Tagebuch
Abstract nouns: die Dankbarkeit
Plurals: Momente
And also many words derived from verbs/adjectives when used as nouns.

So their capitalization here simply shows that they are nouns.