Learn Hardcore Greek: Ο φίλος μου λέει ότι είναι συνηθισμένος στην κίνηση της πόλης, αλλά εγώ όχι ακόμα. - My friend male says that he is used to the traffic of the city, but I am not yet.

So ο is just the normal definite article (masculine singular nominative) that goes with φίλος. You would not normally say φίλος μου alone in this context; ο φίλος μου is the standard way to say my friend as the subject of a sentence.

Why is μου placed after φίλος instead of before it?

Greek possessive pronouns like μου, σου, του, της, μας, σας, τους are clitics and usually follow the noun they modify:

ο φίλος μου = my friend
η αδελφή σου = your sister
το αυτοκίνητό του = his car

So the normal order is: article + noun + possessive clitic.
Putting μου before the noun (μου φίλος) is ungrammatical in modern standard Greek (though you may see fronted possessives in poetry or very marked, emotional speech, and then usually in full form: ο δικός μου φίλος = my own friend).

What exactly does λέει ότι mean here? Could we use πως instead of ότι?

λέει ότι means “(he) says that …” and introduces a reported statement (indirect speech).

λέει ότι είναι συνηθισμένος…
= he says that he is used to…

You can usually replace ότι with πως in modern Greek when introducing reported speech:

Ο φίλος μου λέει ότι είναι συνηθισμένος…
Ο φίλος μου λέει πως είναι συνηθισμένος…

Both mean the same in this sentence and are natural.
Don’t confuse this ότι (conjunction that) with ό,τι (with a comma), which means whatever / anything that. Here it’s the conjunction, so no comma.

Why is it είναι συνηθισμένος and not something like έχει συνηθίσει or συνηθίζει? What’s the difference?

All three forms exist but express slightly different things:

είναι συνηθισμένος σε…
Literally: is accustomed to…
Describes a state: he is already used to it; it’s his current condition.
έχει συνηθίσει σε…
Literally: has gotten used to… (perfect tense of συνηθίζω)
Emphasizes the process + result: he has completed the process of becoming used to it.
συνηθίζει την κίνηση / να οδηγεί…
He gets used to the traffic / to driving …
More like a general or habitual process: he tends (is in the habit) to get used to it, or he is in the process of getting used to it.

In your sentence, είναι συνηθισμένος neatly contrasts with (εγώ) όχι ακόμα:

He is already in the state of being used to it.
I am not yet in that state.

So this choice fits the contrast with ακόμα (yet).

Why is συνηθισμένος masculine? How would it change with a female subject?

συνηθισμένος is a past participle used as an adjective. It must agree in gender, number, and case with the subject.

Subject here: ο φίλος (masculine singular)
→ είναι συνηθισμένος

With a female subject:

Η φίλη μου λέει ότι είναι συνηθισμένη στην κίνηση της πόλης…
(συνηθισμένη, feminine singular)

Plural examples:

Οι φίλοι μου είναι συνηθισμένοι… (masculine/plural or mixed group)
Οι φίλες μου είναι συνηθισμένες… (feminine plural)

So the pattern is:

masc. sg.: συνηθισμένος
fem. sg.: συνηθισμένη
neut. sg.: συνηθισμένο
masc. pl.: συνηθισμένοι
fem. pl.: συνηθισμένες
neut. pl.: συνηθισμένα

What does στην consist of, and why is κίνηση in the accusative?

στην is a contraction of:

σε (preposition in, at, to)
την (feminine singular accusative article the)

So:

στην κίνηση = σε + την κίνηση = in the traffic / to the traffic

The preposition σε in Greek is normally followed by the accusative case, so η κίνηση (nominative) becomes την κίνηση (accusative).

Grammar details:

η κίνηση = the traffic / movement (feminine, nominative)
την κίνηση = accusative form (after σε → στην κίνηση)

What exactly does κίνηση mean here? Is it “movement” or “traffic”?

κίνηση has both meanings, depending on context:

movement, motion
- Η κίνηση του σώματος = the movement of the body
traffic (cars, buses, etc.)
- Υπάρχει πολλή κίνηση στον δρόμο. = There is a lot of traffic on the road.

In your sentence:

στην κίνηση της πόλης

The context (της πόλης, of the city) and everyday usage make κίνηση mean traffic.
So: used to the city traffic.

Why is it της πόλης after κίνηση? Could we say στην κίνηση στην πόλη instead?

της πόλης is a genitive phrase:

της πόλης = of the city
η κίνηση της πόλης = the city’s traffic / the traffic of the city

This is a standard “of-phrase” in Greek using the genitive.

You could say:

στην κίνηση στην πόλη = in the traffic in the city

This is grammatically fine, but slightly different in feel:

η κίνηση της πόλης focuses on the specific city traffic as a characteristic of the city.
η κίνηση στην πόλη focuses more on traffic that happens in the city (more locative).

In most everyday contexts, στην κίνηση της πόλης sounds natural and idiomatic.

What role does αλλά play here? Could we use όμως instead?

αλλά is the basic conjunction meaning “but”:

… αλλά εγώ όχι ακόμα. = … but I (am) not yet.

You could also use όμως (however, though), but it behaves a bit differently:

Ο φίλος μου είναι συνηθισμένος στην κίνηση της πόλης, όμως εγώ όχι ακόμα.

αλλά directly contrasts two clauses, much like English but.
όμως is more like however / though, a bit more flexible in position and slightly more formal or stylistic. Both are correct; αλλά is the straightforward, neutral choice here.

Why is there no verb in αλλά εγώ όχι ακόμα? Shouldn’t it be αλλά εγώ δεν είμαι συνηθισμένος ακόμα?

Greek, like English, often omits repeated verbs when they are clearly understood from the context.

Full form (perfectly correct):

… αλλά εγώ δεν είμαι ακόμα συνηθισμένος (στην κίνηση της πόλης).

Natural shortened form:

… αλλά εγώ όχι ακόμα.
Literally: but I, not yet.

The verb είμαι (am) and the adjective συνηθισμένος (used to) are understood by the listener from the first part of the sentence:

(Ο φίλος μου λέει ότι) είναι συνηθισμένος… αλλά εγώ (δεν είμαι συνηθισμένος) όχι ακόμα.

This kind of ellipsis is very common in spoken and written Greek.

What does ακόμα mean here? Is there a difference between ακόμα and ακόμη, and between όχι ακόμα and ακόμα όχι?

In this sentence, ακόμα means “yet”:

όχι ακόμα = not yet

ακόμα vs ακόμη
In modern standard Greek, ακόμα and ακόμη are usually interchangeable. Many speakers use ακόμα more in speech; ακόμη can feel slightly more formal or literary, but functionally they’re the same in this context.
όχι ακόμα vs ακόμα όχι

όχι ακόμα = not yet (very common):
- Έμαθες να οδηγείς; Όχι ακόμα.
  Have you learned to drive? Not yet.
ακόμα όχι = not yet (but almost / so far no) and often suggests a bit more nuance, like up to now, no:
- Τελείωσες; Ακόμα όχι.
  Finished? Not yet (but I’m working on it).

Here, όχι ακόμα is the most natural, neutral choice.
You also see δεν είμαι ακόμα συνηθισμένος = I am not yet used to… with δεν and ακόμα inside the clause.