Learn Hardcore Greek: Η φιλία μου με μια φίλη μου, η οποία μένει στην Ελλάδα, με βοηθάει να μιλάω περισσότερο. - My friendship with a friend of mine, who lives in Greece, helps me to speak more.

μια φίλη = a (female) friend – could be any friend, no explicit connection to the speaker.
μια φίλη μου = a (female) friend of mine – it shows that this friend belongs to my circle.

So μου after φίλη works like English “of mine”:

έχω μια φίλη = I have a (female) friend.
μιλάω με μια φίλη μου = I’m talking with a (female) friend of mine.

In your sentence, both η φιλία μου and μια φίλη μου have μου, but with slightly different functions:

first μου: possessive on φιλία (my friendship)
second μου: partitive/possessive on φίλη (one of my friends).

Why is μου used twice? Could we avoid repeating it and just say Η φιλία μου με μια φίλη?

You could say Η φιλία μου με μια φίλη, and it would be grammatically correct, but the meaning changes slightly:

με μια φίλη = with a friend (no explicit “of mine”; context might still imply it)
με μια φίλη μου = with a friend of mine (clearly one out of my own friends)

Using μου twice makes both relationships explicit:

η φιλία μου = my friendship
μια φίλη μου = one of my friends

It sounds natural in Greek and is not considered redundant.

What is the role of η οποία in η οποία μένει στην Ελλάδα? How does it compare to που?

η οποία is a relative pronoun meaning roughly “who / which”.
In η φίλη μου, η οποία μένει στην Ελλάδα, it refers back to η φίλη μου and introduces a relative clause:

η οποία μένει στην Ελλάδα = who lives in Greece

Comparison with που:

που is the most common, neutral relative marker:
- η φίλη μου που μένει στην Ελλάδα = my friend who lives in Greece.
η οποία is:
- more formal / careful
- clearly agrees in gender and number with the noun:
  - ο οποίος (masc. sing.)
  - η οποία (fem. sing.)
  - το οποίο (neut. sing.)
  - etc.

In everyday speech, most people would say:

Η φιλία μου με μια φίλη μου, που μένει στην Ελλάδα, …

Your version with η οποία is perfectly correct, just a bit more formal or written style.

Why is there a comma before η οποία? Is that necessary?

Yes, the comma is appropriate here because the clause is non‑defining / non‑restrictive:

μια φίλη μου, η οποία μένει στην Ελλάδα, …

This implies:

You already know which friend we’re talking about (from context),
And η οποία μένει στην Ελλάδα gives extra information about her, not information that identifies which friend.

In Greek, as in English, non‑defining relative clauses are usually set off with commas:

Η Μαρία, η οποία είναι γιατρός, μένει εδώ κοντά.
Maria, who is a doctor, lives nearby.

If the clause were defining (needed to identify the person), you’d usually skip the commas and tend to use που:

Μια φίλη μου που μένει στην Ελλάδα με βοηθάει…
A friend of mine who lives in Greece helps me… (specifically that friend, not another).

What’s the meaning and nuance of μένει in η οποία μένει στην Ελλάδα? Could we use ζει instead?

μένω = to stay / live / reside (somewhere)
ζω = to live (in the sense of being alive, or living one’s life; but also commonly used as “reside”)

In this sentence:

η οποία μένει στην Ελλάδα ≈ who lives in Greece, with a slightly stronger feeling of residing / staying there.
You could also say η οποία ζει στην Ελλάδα with practically the same meaning; it’s perfectly natural.

Nuance:

μένω στην Ελλάδα = I live / reside in Greece (very common, everyday way to say where you live).
ζω στην Ελλάδα = also “I live in Greece”, but ζω can carry more existential connotations in other contexts (e.g. ζω μια ήσυχη ζωή = I live a quiet life).

In με βοηθάει, what is the function of με, and is βοηθάει different from βοηθά?

με here is the weak object pronoun: me.
- με βοηθάει = she helps me / it helps me.
About βοηθάει vs βοηθά:
- The verb is βοηθάω / βοηθώ = to help.
- In the 3rd person singular, you commonly see:
  - με βοηθάει
  - με βοηθά
- Both are correct and mean the same thing: he/she/it helps me.
- βοηθάει (with -άει) is a bit more “spelled out” and common in informal writing.
- βοηθά (without -ει) is a bit more concise and can feel slightly more standard/neutral.

So:

Η φιλία μου … με βοηθάει να μιλάω περισσότερο.
Η φιλία μου … με βοηθά να μιλάω περισσότερο.

Both are fine.

Why do we use να μιλάω instead of να μιλήσω? What is the difference?

The difference is aspect:

μιλάω (imperfective aspect) = to be speaking / to speak in general, habitually, repeatedly.
μιλήσω (aorist aspect) = to speak once, to manage to speak, to get a chance to speak, to complete a speaking event.

With να (the particle introducing a subjunctive-like clause):

να μιλάω περισσότερο = to speak more / to speak more often / to spend more time speaking (ongoing or repeated action).
να μιλήσω περισσότερο would sound more like to speak more (on that specific occasion / to say more things).

In your sentence, the idea is that this friendship helps you speak Greek more often / more in general, so να μιλάω (imperfective) is the natural choice.

Why is it μιλάω and not μιλώ? Are both correct?

Yes, both are correct forms of the same verb:

μιλάω
μιλώ

They both mean “I speak / I talk”. In modern Greek:

The -άω form (μιλάω) is very common in everyday speech.
The -ώ form (μιλώ) is a bit shorter, sometimes sounds slightly more formal or just more “standard” on paper.

Examples:

Μιλάω ελληνικά. / Μιλώ ελληνικά. = I speak Greek.

So you could also say:

…με βοηθάει να μιλώ περισσότερο.
It’s correct, just slightly different stylistically. Using μιλάω is perfectly natural.

Why do we say στην Ελλάδα and not σε την Ελλάδα?

σε + την contracts to στην in spoken and written Greek.

σε = in / at / to
την Ελλάδα = the Greece (fem. sing. accusative)
→ σε την Ελλάδα → στην Ελλάδα

This contraction happens very regularly:

σε + την → στην
σε + τη → στη
σε + τον → στον
σε + το → στο

So:

μένω στην Ελλάδα = I live in Greece.
πάω στο σχολείο = I go to school.

Could we change the word order, for example say Η φιλία μου με μια φίλη μου με βοηθάει vs … με βοηθάει η φιλία μου? How flexible is the word order here?

Greek word order is relatively flexible, but there are preferences:

Clitic pronoun position
The object pronoun με normally goes before the verb in the indicative:
- με βοηθάει (correct, standard)
- βοηθάει με (only in special emphatic or poetic contexts; usually avoided in normal speech).
Subject vs verb:
- Η φιλία μου … με βοηθάει (subject–verb) is perfectly natural.
- … με βοηθάει η φιλία μου is also grammatical; it might give some extra emphasis to η φιλία μου at the end.

In your full sentence, the most natural-sounding version is exactly as given:

Η φιλία μου με μια φίλη μου, η οποία μένει στην Ελλάδα, με βοηθάει να μιλάω περισσότερο.

You could reorder slightly, but you should keep με right before βοηθάει:

Η φιλία μου με μια φίλη μου, η οποία μένει στην Ελλάδα, με βοηθά να μιλάω περισσότερο. (same meaning, slightly shorter verb form)

Is there any difference between μια φίλη μου and μία φίλη μου in this sentence?

In everyday pronunciation, μια and μία sound the same. The difference is mostly orthographic and sometimes about emphasis:

μια φίλη μου is the usual spelling for the indefinite article “a / one” before a feminine noun.
μία φίλη μου (with the accented ί) is often used when you really mean “one (single) friend of mine” with a contrast or emphasis on the number.

In your sentence:

Η φιλία μου με μια φίλη μου…
is the normal, neutral way to write it.

If you wrote:

Η φιλία μου με μία φίλη μου…
it could (depending on context) slightly suggest “with one friend of mine (as opposed to several)”.