Τον Νοέμβριο και τον Δεκέμβριο προτιμώ να μένω σπίτι με ένα βιβλίο και ζεστό τσάι.

Because the sentence uses the accusative case for a time expression.

The dictionary forms are:

• ο Νοέμβριος
• ο Δεκέμβριος

But when Greek says in November or in December, it commonly uses:

• τον Νοέμβριο
• τον Δεκέμβριο

So the ending changes because the nouns are not in the nominative anymore.

Word-for-word, yes, it contains the: τον = masculine singular accusative the.

But you should not translate it literally. In this kind of expression, τον + month in the accusative means in + month in natural English.

So:

• τον Νοέμβριο = in November
• τον Δεκέμβριο = in December

This is just a normal Greek way of expressing time.

Because each month keeps its own article:

• τον Νοέμβριο και τον Δεκέμβριο

Repeating the article is the most natural and standard way here. Greek often does this with coordinated nouns, especially when each one is being clearly identified.

Because Modern Greek usually does not use an infinitive the way English does.

English says:

• I prefer to stay

Greek says:

• προτιμώ να μένω

Here να introduces the following verb. In many cases, να + verb does the job that to + verb does in English.

So:

• προτιμώ να μένω = I prefer to stay

This is about aspect, which is very important in Greek.

• να μένω = imperfective, suggesting something repeated, ongoing, or habitual
• να μείνω = perfective, suggesting a single complete event

In this sentence, the speaker means a general preference during November and December, not one single occasion. So να μένω is the natural choice.

Compare:

• Τον χειμώνα προτιμώ να μένω σπίτι.
  In winter I prefer to stay home.
  Habitual/general

• Απόψε προτιμώ να μείνω σπίτι.
  Tonight I prefer to stay home.
  One specific occasion

Because σπίτι can work almost like an adverbial expression meaning home.

So:

• μένω σπίτι = I stay home / I stay at home

This is very natural Greek.

If you say:

• μένω στο σπίτι

that is also correct, but it sounds a bit more literal: I stay in the house / at the house / at home. It can feel slightly more specific or concrete.

So in this sentence, bare σπίτι fits the general, cozy, habitual idea very well.

Both can mean I stay at home, but there is a slight nuance:

• μένω σπίτι = more idiomatic, more like stay home
• μένω στο σπίτι = a little more explicit, more like stay in the house / remain at home

In everyday speech, both are common. In this sentence, μένω σπίτι sounds very natural because it expresses a general preference, not focus on a specific building.

Because με in Modern Greek is followed by the accusative case.

So after με, the nouns are in the accusative:

• με ένα βιβλίο
• με ζεστό τσάι

In this sentence, both βιβλίο and τσάι are neuter nouns, and neuter nominative and accusative are often identical in form, so the words do not visibly change much.

Also notice agreement:

• ένα βιβλίο: ένα matches the neuter singular noun βιβλίο
• ζεστό τσάι: ζεστό matches the neuter singular noun τσάι

Because βιβλίο is a countable noun, while τσάι is often treated as a mass noun.

So:

• με ένα βιβλίο = with a book
• με ζεστό τσάι = with warm tea

Greek often leaves out the indefinite article with mass nouns like tea, water, coffee, especially when speaking generally.

You could also say:

• με ένα βιβλίο και ένα ζεστό τσάι

and that would be grammatical too. It just sounds a little more explicitly like a warm tea as one item, while ζεστό τσάι feels slightly more general and natural.

No, Greek word order is fairly flexible.

The original sentence starts with the time phrase:

• Τον Νοέμβριο και τον Δεκέμβριο ...

This sets the time frame first, which is very natural.

You could rearrange parts of the sentence, for example:

• Προτιμώ τον Νοέμβριο και τον Δεκέμβριο να μένω σπίτι...

But the original version sounds smooth and natural because it first tells you when, then what the speaker prefers.

So the order is not fixed, but the given order is a very good, idiomatic choice.