Learn Hardcore Greek: Στον καναπέ στο σαλόνι κάθομαι άνετα. - On the sofa in the living room I sit comfortably.

στον καναπέ = σε
- τον καναπέ
  - καναπές is masculine, so its article in the accusative is τον → σε + τον = στον
στο σαλόνι = σε
- το σαλόνι
  - σαλόνι is neuter, so its article in the accusative is το → σε + το = στο

So the ν is there because καναπές is masculine and needs τον.
Neuter nouns use το, which contracts to στο (no ν).

What case are καναπέ and σαλόνι, and why are they in that case?

Both καναπέ and σαλόνι are in the accusative case:

στον καναπέ = σε
- accusative
στο σαλόνι = σε
- accusative

In modern Greek, the preposition σε (in, on, at, to) normally takes the accusative. So for location:

στον καναπέ = on the couch
στο σαλόνι = in the living room

Even though in older stages of Greek location could take a different case, in modern Greek σε + accusative is standard.

What exactly does στον / στο mean? Is it “in”, “on”, or “at”?

σε is a very flexible preposition that often corresponds to English in, on, or at, depending on context.

στον καναπέ → literally on the couch
στο σαλόνι → literally in the living room

You don’t usually need a separate word for in vs on; Greek just uses σε plus the noun, and context (and the nature of the noun) tells you if it feels like in, on, or at.

Why is the subject “I” not written? Where is the “I” in this sentence?

Greek usually drops subject pronouns because the verb ending shows the person.

The verb κάθομαι is 1st person singular (I sit / I am sitting).
So:

κάθομαι = I sit / I am sitting

You can say Εγώ κάθομαι άνετα… if you want to emphasize I (as opposed to someone else), but normally just κάθομαι is enough and sounds more natural.

What kind of verb is κάθομαι and why does it end in -ομαι?

κάθομαι is a middle/passive form in modern Greek, but it is active in meaning, like an English reflexive or intransitive:

κάθομαι = I sit (myself down) / I am sitting

Many common verbs in Greek use this -ομαι form but are translated with simple active verbs in English, e.g.:

ξαπλώνω / ξαπλώνομαι – I lie down
σκεφτομαι – I think

So even though κάθομαι looks like a passive, you simply learn it as I sit.

What is the difference between κάθομαι and κάτσω or κάθισα?

These are different aspects/tenses of the verb “to sit”:

κάθομαι – present, I sit / I am sitting (state or ongoing action)
κάθισα – aorist, I sat (down) (completed action in the past)
θα κάτσω – future, I will sit (down) or I’ll take a seat

In your sentence, κάθομαι describes the current state:
Στον καναπέ στο σαλόνι κάθομαι άνετα.
= I am sitting comfortably on the couch in the living room.

Why is it άνετα and not άνετος?

άνετος is the adjective (masculine form): comfortable (describing a noun).
άνετα is the adverb: comfortably (describing how you do something).

άνετος καναπές = a comfortable couch (adjective)
κάθομαι άνετα = I sit comfortably (adverb)

Since you are describing how you sit, you need the adverb άνετα, not the adjective άνετος.

Can άνετα go somewhere else in the sentence, or must it come at the end?

It can move, but the end position is very natural and common:

Στον καναπέ στο σαλόνι κάθομαι άνετα. (neutral, natural)
Στον καναπέ στο σαλόνι άνετα κάθομαι. (possible, a bit more emphatic on άνετα)
Άνετα κάθομαι στον καναπέ στο σαλόνι. (emphasizes comfortably even more)

Word order in Greek is flexible, and moving άνετα generally changes emphasis, not basic meaning.

Is the word order fixed? Could I say Κάθομαι άνετα στον καναπέ στο σαλόνι instead?

Yes, you can. All of these are grammatically fine and natural:

Στον καναπέ στο σαλόνι κάθομαι άνετα.
Κάθομαι άνετα στον καναπέ στο σαλόνι.
Κάθομαι στον καναπέ στο σαλόνι άνετα.

Greek allows more flexibility in word order than English. Changing the order mostly affects rhythm and emphasis, not the core meaning.

The original version gives a bit of focus to the place first, and then what happens there.

Does στον καναπέ στο σαλόνι mean “on the couch that is in the living room”, or “in the living room, on the couch”? Is there a difference?

In practice, it’s understood as:

You are in the living room
You are sitting on the couch there
You are comfortable

So effectively: I sit comfortably on the couch in the living room.

Both of your English paraphrases amount to the same thing here. Greek doesn’t strictly mark whether στο σαλόνι modifies κάθομαι or καναπέ; context makes it clear. Listeners will just imagine the couch in the living room.

Do I really need both στον καναπέ and στο σαλόνι? Could I just say one of them?

You can definitely shorten it:

Στον καναπέ κάθομαι άνετα. = On the couch I sit comfortably.
Στο σαλόνι κάθομαι άνετα. = In the living room I sit comfortably.

The full sentence simply gives more specific location:
not just any couch, but the couch in the living room.

Could I say στον καναπέ του σαλονιού instead of στον καναπέ στο σαλόνι? What’s the difference?

Yes:

στον καναπέ του σαλονιού = on the living-room couch (literally: on the couch of the living room)

Difference in feel:

στον καναπέ στο σαλόνι – sounds like you’re just describing where you are: on the couch in the living room.
στον καναπέ του σαλονιού – treats the living-room couch more like a single fixed object (maybe there’s also a couch in another room).

Both are correct; the original is slightly more neutral and everyday.

In English I would say “I am sitting comfortably…”. Is Greek κάθομαι more like “I sit” or “I am sitting”?

Greek present usually covers both English simple present and present continuous.

κάθομαι can be:
- I sit (habitual/repeated)
- I am sitting (right now)

Context decides. In your sentence, it is most natural to understand it as I am sitting comfortably on the couch in the living room (current situation).