Learn Hardcore Greek: Ο φίλος μου είναι πολύ οργανωμένος στη δουλειά.

Putting μου in front (μου φίλος) is wrong in modern Greek (except in very special, poetic or dialectal contexts), so you should learn:
article + noun + possessive pronoun as the normal order.

Do I always need the article Ο in Ο φίλος μου, or can I just say Φίλος μου είναι…?

Most of the time, when you talk about a specific person (“my friend” as a known individual), Greek uses the definite article:

Ο φίλος μου είναι… = My friend is…

You can sometimes drop the article:

Φίλος μου είναι…

…but that usually sounds either:

more stylistic / literary, or
like you’re saying “a friend of mine” rather than clearly “my (particular) friend”.

For everyday speech, Ο φίλος μου is by far the most natural and common choice.

What exactly is μου here? Is it a separate word, and why doesn’t it have an accent?

Μου is the unstressed (clitic) possessive pronoun meaning my.

It is a separate word, but it is pronounced together with the noun before it.
Because it is unstressed, it normally does not carry an accent in writing.

Examples:

ο φίλος μου = my friend
το σπίτι μου = my house
η δουλειά μου = my job

There is also a stressed form used for emphasis, with δικός etc.:

ο δικός μου φίλος = my friend (as opposed to someone else’s)

But in neutral sentences like this one, μου without accent after the noun is the standard possessive form.

What form of the verb is είναι? Where is the word “he” in Greek?

Είναι is the 3rd person singular of the verb είμαι (to be):

είμαι = I am
είσαι = you (sg.) are
είναι = he / she / it is

Greek usually drops subject pronouns (I, you, he, etc.) because the verb ending already shows the person.

So:

(Αυτός) είναι πολύ οργανωμένος.
= He is very organized.

Αυτός (he) is optional. In your sentence, Ο φίλος μου is the subject, so you don’t need αυτός as well:

Ο φίλος μου είναι… = My friend is…

Why does οργανωμένος end in -ος? Does it agree with φίλος?

Yes. Οργανωμένος is an adjective (originally a past participle) meaning organized, and in Greek adjectives must agree with the noun in:

gender (masculine / feminine / neuter)
number (singular / plural)
case (nominative / accusative / etc.)

Here:

ο φίλος = masculine, singular, nominative
So the adjective must also be masculine, singular, nominative → οργανωμένος

Other forms of the same adjective:

Masculine: οργανωμένος (sg.), οργανωμένοι (pl.)
Feminine: οργανωμένη (sg.), οργανωμένες (pl.)
Neuter: οργανωμένο (sg.), οργανωμένα (pl.)

So if the friend were female:

Η φίλη μου είναι πολύ οργανωμένη στη δουλειά.
= My (female) friend is very organized at work.

What is the role of πολύ here? Is it an adjective or an adverb?

In είναι πολύ οργανωμένος, πολύ functions as an adverb meaning very.

It modifies the adjective οργανωμένος:
- οργανωμένος = organized
- πολύ οργανωμένος = very organized

As an adverb, πολύ:

does not change for gender/number/case
usually comes before the adjective or adverb it modifies

Compare:

είναι οργανωμένος = he is organized
είναι πολύ οργανωμένος = he is very organized

There are also adjective forms (with gender/number):

πολύς (masc.), πολλή (fem.), πολύ (neut.) = much / a lot of
- πολλή δουλειά = a lot of work
- πολύς κόσμος = a lot of people

So:

πολύ οργανωμένος → πολύ = adverb (very)
πολλή δουλειά → πολλή = adjective (much / a lot of)

Could I move πολύ after the adjective, like είναι οργανωμένος πολύ?

You can say είναι οργανωμένος πολύ, and it’s grammatically correct, but:

πολύ before the adjective (είναι πολύ οργανωμένος) is the most natural and neutral order.
πολύ after the adjective (είναι οργανωμένος πολύ) often sounds:
- more emphatic, or
- more colloquial / expressive, depending on intonation.

So for standard usage, especially as a learner, prefer:

είναι πολύ οργανωμένος.

What exactly is στη? Is it one word, and why not σε τη δουλειά?

Στη is a contraction of:

σε (preposition: in / at / to)
τη(ν) (feminine singular definite article: the)

So:

σε + τη δουλειά → στη δουλειά

In modern Greek, σε + article is almost always written and pronounced as one word:

σε + τον → στον (e.g. στον δρόμο = in/on the street)
σε + το → στο (e.g. στο σπίτι = at home / in the house)
σε + την → στη(ν) (e.g. στη δουλειά = at work)

So you should write and say στη δουλειά, not σε τη δουλειά.

Why is it sometimes στη δουλειά and sometimes στην δουλειά? Is one of them wrong?

Both στη δουλειά and στην δουλειά are used.

Historically and formally: στην (with ν) is the full form:
- σε + την δουλειά → στην δουλειά
In modern everyday speech and writing, the ν is often dropped before certain consonants, especially δ, θ, κ, π, τ, γ, χ, φ, σ, ζ.

So:

στη δουλειά (without ν) is very common and perfectly correct.
στην δουλειά (with ν) is also correct, just a bit more careful / traditional spelling.

As a learner, you will see both; στη δουλειά is probably what you’ll hear most often in speech.

Does στη δουλειά mean “in the work” or “at work”? Why use σε here?

Σε is a very flexible preposition in Greek, covering meanings of in, at, and to, depending on the context.

In στη δουλειά:

Literally: in/at the work
Natural English translation: at work

Typical uses:

είμαι στη δουλειά = I am at work
πάω στη δουλειά = I’m going to work
μένω στο σπίτι = I live at home / in the house
είμαι στην Αθήνα = I am in Athens

So σε + article (here στη) is the standard way to say at work in Greek.

Is there a difference between δουλειά and εργασία for “work / job”?

Yes, there is a difference in register and nuance:

δουλειά
- most common in everyday speech
- can mean work in general or a job
- used in informal and neutral contexts
- e.g. Έχω πολλή δουλειά. = I have a lot of work.
εργασία
- more formal and bookish
- used in official, academic, or technical contexts
- can mean work, employment, or an assignment / project
- e.g. γραφική εργασία, πτυχιακή εργασία

In your sentence, στη δουλειά is the natural, colloquial way to say at work.
Στην εργασία is possible, but would sound quite formal in this everyday context.