Τον Σεπτέμβριο και τον Οκτώβριο εγώ φοράω ακόμα μαύρα παπούτσια, αλλά η αδερφή μου θέλει ήδη μπότες.

Because Greek often uses the accusative for expressions of time meaning during / in a certain month or period.

So:

• τον Σεπτέμβριο = in September / during September
• τον Οκτώβριο = in October / during October

The article τον shows the masculine accusative singular, because Σεπτέμβριος and Οκτώβριος are masculine nouns.

This is very common in Greek with months:

• Τον Ιανουάριο πάω διακοπές.
  = I go on holiday in January.

Greek usually repeats the article when mentioning two separate nouns, even if they are linked by και.

So:

• τον Σεπτέμβριο και τον Οκτώβριο

sounds natural and standard.

You may sometimes see one article covering both items in other contexts, but here repeating it is the normal way to say in September and in October. It makes each month feel like its own full time expression.

Yes. The ending of φοράω already tells you the subject is I, so εγώ is not required.

It is included here for emphasis or contrast:

• εγώ φοράω ακόμα μαύρα παπούτσια, αλλά η αδερφή μου...

So the idea is something like:

• I still wear black shoes, but my sister already wants boots.

In Greek, subject pronouns are often added when the speaker wants to stress the contrast.

They are two forms of the same verb, meaning to wear.

• φοράω
• φορώ

Both are correct in Modern Greek. Φοράω is very common in everyday speech and writing. Φορώ is also standard and common.

So these are both possible:

• Εγώ φοράω μαύρα παπούτσια.
• Εγώ φορώ μαύρα παπούτσια.

Same meaning.

Because the adjective has to agree with the noun in gender, number, and case.

Here:

• παπούτσια = shoes
• it is neuter plural
• so the adjective must also be neuter plural

That gives:

• μαύρα παπούτσια = black shoes

Compare:

• μαύρο παπούτσι = black shoe
• μαύρα παπούτσια = black shoes

Because Greek often leaves out the article when speaking in a more general or indefinite way.

So:

• θέλει ήδη μπότες

means something like:

• she already wants boots
• she already wants some boots

It does not mean a specific previously mentioned pair of boots.

If you said τις μπότες, that would sound more like the boots, referring to specific boots already known from the context.

In Greek, family words and many other nouns often appear with the definite article, even when they are followed by a possessive like μου.

So Greek naturally says:

• η αδερφή μου = my sister
• ο αδερφός μου = my brother
• η μητέρα μου = my mother

This is one of the places where Greek works differently from English. In English, we do not say the my sister, but in Greek the article is normal.

Yes. They both mean sister.

• αδερφή is the very common modern everyday form
• αδελφή is more conservative/formal and closer to the older form

A native speaker will understand both. In everyday modern Greek, αδερφή is extremely common.

The same applies to:

• αδερφός / αδελφός = brother

They are time-related adverbs:

• ακόμα = still
• ήδη = already

So:

• φοράω ακόμα μαύρα παπούτσια = I still wear black shoes
• θέλει ήδη μπότες = she already wants boots

They help create the contrast:

• I am still in the shoes stage
• my sister is already at the boots stage

Yes, Greek word order is more flexible than English word order.

This sentence is arranged to sound natural and to highlight the contrast:

• Τον Σεπτέμβριο και τον Οκτώβριο εγώ φοράω ακόμα μαύρα παπούτσια, αλλά η αδερφή μου θέλει ήδη μπότες.

The order puts the time expression first, then emphasizes εγώ, and then contrasts that with η αδερφή μου.

Other orders are possible, for example:

• Εγώ φοράω ακόμα μαύρα παπούτσια τον Σεπτέμβριο και τον Οκτώβριο...

That is grammatically fine, but it may sound slightly different in focus. Greek often moves parts around to emphasize what matters most.

It can suggest either, depending on context.

Literally:

• θέλει μπότες = she wants boots

But in a sentence like this, it often implies the more natural idea:

• she already wants to start wearing boots
• she already feels ready for boots / boot weather

So the Greek is literal, but the real sense in context may be broader than simply desiring to own boots.