Learn Hardcore Greek: Η αδερφή μου με συγχωρεί όταν της ζητάω συγγνώμη με ειλικρίνεια. - My sister forgives me when I apologize to her sincerely.

You can drop the article (Αδερφή μου, έλα εδώ = Sister, come here), but then it usually sounds like direct address (calling someone) or has a more emotional/poetic tone. In neutral statements, keep the article: Η αδερφή μου μένει στην Αθήνα.

What is μου in Η αδερφή μου exactly? Is it like my or me?

Μου here is a weak (clitic) pronoun in the genitive case, functioning as a possessive: my.
So:

εγώ = I
μου = of me → my

With nouns, μου usually comes after the noun: η αδερφή μου (not μου αδερφή). Grammar-wise it's closer to of me, but in meaning it’s just my.

Why is με before the verb in με συγχωρεί? In English it’s forgives me, after the verb.

Με is the weak form of εμένα (me) in the accusative case and it’s a clitic object pronoun.
In standard Greek word order, weak object pronouns go before a simple present/past verb:

με συγχωρεί = she forgives me
τον βλέπω = I see him
σε αγαπώ = I love you

You would only put the full form εμένα after the verb for emphasis: Συγχωρεί εμένα, όχι εσένα. (She forgives me, not you.)

What’s the difference between αδερφή and αδελφή?

They are the same word: sister.

αδελφή is the more traditional / formal spelling.
αδερφή reflects the modern pronunciation and is very common in everyday writing.

Pronunciation is essentially the same: a-ther-FEE. Both are correct; in informal contexts you’ll often see αδερφή.

What tense is συγχωρεί and why is it used here?

Συγχωρεί is the present tense, active voice, 3rd person singular of συγχωρώ (I forgive).
Greek present tense covers both English simple present and present continuous.

Here, with όταν (when), it expresses a general, repeated situation:
Η αδερφή μου με συγχωρεί όταν… = My sister forgives me when(ever)…

Why is it της ζητάω and not something like ζητάω σε αυτήν for I ask her?

In Greek, the indirect object (the person you are asking) is usually in the genitive case using a pronoun:

της ζητάω = I ask her / I ask of her
του μιλάω = I talk to him

So της here is to her / from her, built into the verb phrase.
You can say ζητάω συγγνώμη από αυτήν (I ask forgiveness from her), but της ζητάω συγγνώμη is shorter and more natural in speech.

Why is it της ζητάω συγγνώμη and not της ζητάω μια συγγνώμη?

Συγγνώμη in this meaning behaves like an abstract noun or set phrase, closer to apology / forgiveness in general, not a countable “one apology”.
So the usual pattern is:

ζητάω συγγνώμη = I apologize / I ask for forgiveness

Adding an article (μια συγγνώμη) is possible in some contexts, but it sounds more like “an apology” as a specific instance and is less common in this fixed expression.

Isn’t it redundant to have both συγχωρεί (forgives) and συγγνώμη (sorry/forgiveness) in the same sentence?

It may look redundant from English, but in Greek it’s a very normal combination.
The logic is:

με συγχωρεί = she forgives me
όταν της ζητάω συγγνώμη = when I apologize / when I ask her for forgiveness

So you’re saying: She forgives me when I apologize to her sincerely.
You’re not repeating the same word twice in Greek; συγχωρεί and συγγνώμη are related but different words.

What’s the difference between ζητάω and ζητώ?

They are two forms of the same verb (ζητάω / ζητώ = I ask / I request).

ζητάω is more colloquial, very common in everyday speech.
ζητώ is a bit shorter and slightly more formal/literary.

In modern Greek, you can usually use them interchangeably; ζητάω συγγνώμη and ζητώ συγγνώμη are both correct.

Is συγγνώμη always spelled with double γγ? I sometimes see συγνώμη.

The standard spelling is συγγνώμη with γγ.
However, συγνώμη (with one γ) is very common in informal writing (texts, social media, etc.) and widely understood.

If you want to be on the safe side for correct spelling (exams, formal writing), use συγγνώμη.

Why is it με ειλικρίνεια and not ειλικρινά? Are they the same?

They are very close in meaning:

με ειλικρίνεια = with sincerity, literally a prepositional phrase (με
- noun)
ειλικρινά = sincerely / honestly, an adverb

Both fit here:

…όταν της ζητάω συγγνώμη με ειλικρίνεια.
…όταν της ζητάω συγγνώμη ειλικρινά.

Ειλικρινά sounds a bit more direct and conversational; με ειλικρίνεια feels a little more formal or careful.

Can the word order change, for example: Η αδερφή μου συγχωρεί με όταν της ζητάω συγγνώμη?

You cannot normally say συγχωρεί με with the weak pronoun με after the verb; that sounds wrong in standard Greek.
The clitic pronoun has a fixed position:

Before a simple verb: με συγχωρεί
Between auxiliary and participle, or between particle and verb in more complex forms.

What you can move around more freely are longer phrases:

Η αδερφή μου με συγχωρεί όταν της ζητάω συγγνώμη.
Όταν της ζητάω συγγνώμη, η αδερφή μου με συγχωρεί.

Both are correct; the first is more neutral.

Why is όταν followed by the present tense (ζητάω) and not a future or subjunctive form?

With όταν meaning whenever / when (repeatedly), Greek normally uses the present tense to describe general or habitual events:

Όταν πεινάω, τρώω. = When(ever) I’m hungry, I eat.
Όταν της ζητάω συγγνώμη, με συγχωρεί. = When(ever) I apologize to her, she forgives me.

For a single future event, you would see όταν + subjunctive with θα or να in some constructions, but here the meaning is clearly habitual, so simple present is correct.

How are the object pronouns working together in με συγχωρεί όταν της ζητάω συγγνώμη?

You have two different object pronouns, each with a different function:

με (accusative) = direct object of συγχωρεί → she forgives me
της (genitive) = indirect object of ζητάω → I ask her / I ask of her

Structurally:

Η αδερφή μου (subject)
με συγχωρεί (verb + direct object)
όταν της ζητάω συγγνώμη (subordinate clause with verb + indirect object + noun)

Both pronouns are weak (clitic) forms and come right next to their verbs.