Η σερβιτόρα άφησε τον δίσκο πάνω στον πάγκο, κι εγώ πήρα την κούπα μου και κάθισα στο σκαμπό.

They are all in the aorist, which is the normal Greek tense for a completed past action.

In this sentence, the aorist gives a clear sequence of events:

• the waitress set down the tray
• I took my mug
• I sat down on the stool

So it works much like a simple narrative past in English.

Yes. Greek verb endings already show the subject, so πήρα by itself already means I took.

Adding εγώ gives emphasis or contrast. Here it sounds like:

• and I took my mug...
• as for me, I took my mug...

So εγώ is not grammatically necessary, but it is very natural when the speaker wants to make the subject more explicit.

κι is a shortened form of και.

It is very common before a vowel, so κι εγώ is just a smoother, more natural way to say και εγώ. Both mean and I.

So:

• και εγώ = full form
• κι εγώ = shortened, very common form

There is no important difference in meaning here.

In Greek, unstressed possessive pronouns like μου, σου, του, της usually come after the noun.

So:

• η κούπα μου = my mug
• το βιβλίο σου = your book

Greek normally keeps the article too, so you do not usually say just κούπα μου in a neutral sentence like this. The normal pattern is:

article + noun + possessive pronoun

So:

• την κούπα μου = my mug in the accusative

Because these nouns are direct objects, so they are in the accusative case.

Compare:

• ο δίσκος = the tray (nominative, subject form)

• τον δίσκο = the tray (accusative, object form)

• η κούπα = the mug (nominative)
• την κούπα = the mug (accusative)

In the sentence:

• Η σερβιτόρα is the subject, so it stays nominative.
• τον δίσκο and την κούπα are objects, so they are accusative.

These are contractions of σε + article:

• σε τον → στον
• σε το → στο

So:

• στον πάγκο = on/at the counter
• στο σκαμπό = on the stool

This contraction is completely standard in modern Greek.

Also, after σε, Greek uses the accusative:

• ο πάγκος → τον πάγκο
• το σκαμπό → το σκαμπό (same form here because it is neuter)

Literally, it means on top of the counter.

The word πάνω adds the idea of being on the surface of something. So:

• πάνω στον πάγκο = on the counter / on top of the counter

Without πάνω, στον πάγκο could sometimes be understood more generally as at the counter or on the counter, depending on context.
Using πάνω makes the physical placement clearer.

Because παίρνω has an irregular aorist.

Its basic forms are:

• παίρνω = I take
• πήρα = I took

This is very common in Greek: some verbs do not form the aorist in a fully predictable way, so learners usually memorize the aorist as an important principal form.

So here:

• πήρα την κούπα μου = I took my mug

Because κάθομαι has the aorist form κάθισα.

This is a very common pattern in Greek: the present form and the aorist form can look quite different.

• κάθομαι = I sit / I am sitting
• κάθισα = I sat down / I sat

In this sentence, κάθισα refers to the completed action of sitting down.

So κάθισα στο σκαμπό means:

• I sat on the stool
• often more naturally, I sat down on the stool

The verb αφήνω can mean different things depending on context, including:

• leave
• let
• put down / set down

Here, because the object is τον δίσκο and the location is πάνω στον πάγκο, the natural meaning is:

• set the tray down
• put the tray on the counter

So the context tells you which English translation fits best.

Greek uses the definite article very regularly with specific nouns, often more consistently than English learners expect.

In this sentence, all of these refer to identifiable things in the situation:

• the waitress
• the tray
• the mug
• the counter
• the stool

Also, with possessives, Greek normally still keeps the article:

• την κούπα μου = my mug

So the heavy use of articles here is completely normal Greek.

Greek word order is more flexible than English, but this order is very natural for storytelling.

This sentence follows a smooth narrative flow:

1. what the waitress did
2. what I did next
3. the final action

You could change the order for emphasis, but the given version sounds neutral and natural. Greek often uses word order to highlight what is most important, while the case endings and verb endings still show the grammatical roles.