Τον Δεκέμβριο μένω συχνά σπίτι το βράδυ, γιατί κάνει πολύ κρύο.

Because Greek often uses the accusative to express time, especially with months, days, and parts of the day.
So Τον Δεκέμβριο means in December / during December.

The article τον is normal here. It does not mean the December in an English sense. It is just part of the Greek time expression.

For example:

• Τον Ιανουάριο = in January
• Την Κυριακή = on Sunday
• Το πρωί = in the morning

Δεκέμβριο is in the accusative singular. The basic dictionary form is Δεκέμβριος.

Masculine nouns like this often change:

• nominative: ο Δεκέμβριος
• accusative: τον Δεκέμβριο

It changes because the phrase is functioning as a time expression, and Greek usually uses the accusative for that.

Μένω can mean different things depending on context:

• I live
• I stay
• I remain

Here it means I stay. In this sentence, the idea is not where you permanently live, but that you often stay at home in the evenings during December.

So μένω σπίτι here is closer to I stay home than I live at home.

Greek often uses σπίτι by itself in an idiomatic way to mean home / at home.

So:

• μένω σπίτι = I stay home
• πάω σπίτι = I go home

This is similar to English, where we often say go home rather than go to home.

You can also hear στο σπίτι, but that sounds more like in the house / at the house, with a little more focus on the physical location.
By contrast, bare σπίτι often has the more natural, idiomatic sense of home.

Το βράδυ is another accusative time expression. It means in the evening / at night, depending on context.

Greek very often uses:

• το πρωί = in the morning
• το μεσημέρι = at noon / in the midday period
• το απόγευμα = in the afternoon
• το βράδυ = in the evening / at night

So the article is normal here too.

Yes, usually there is a slight difference.

• το βράδυ usually refers to the evening or the earlier part of the night
• τη νύχτα usually means during the night, often later and darker hours

So in this sentence, το βράδυ sounds natural if the speaker means they stay home in the evenings.

Yes, συχνά can move. Greek word order is fairly flexible.

Here:

• μένω συχνά σπίτι = I often stay home

You could also hear:

• συχνά μένω σπίτι
• μένω σπίτι συχνά

The basic meaning stays the same, but the emphasis can shift slightly.
The original sentence sounds very natural and neutral.

Yes. Γιατί can mean both:

• why in a question
• because in an answer or explanation

Here it means because:

• γιατί κάνει πολύ κρύο = because it is very cold

This is very common in everyday Greek.
Another word you may see for because is επειδή, but γιατί is extremely common in speech.

It is an impersonal weather expression. Greek often uses κάνει with weather conditions:

• κάνει κρύο = it is cold
• κάνει ζέστη = it is hot
• κάνει καλό καιρό = the weather is good

Literally, this does not match English word-for-word, but it is completely natural Greek.
So κάνει πολύ κρύο simply means it is very cold.

Because the fixed expression is κάνει κρύο. Here κρύο is the form used in that impersonal expression, and πολύ means very / a lot of.

So:

• κάνει κρύο = it is cold
• κάνει πολύ κρύο = it is very cold

You do not use κρύος here, because that form is an adjective meaning cold when describing a masculine noun, such as:

• κρύος καφές = cold coffee

Both are in the present tense.

In this sentence, the present tense expresses a habitual action:

• μένω συχνά σπίτι = I often stay home
• κάνει πολύ κρύο = it is very cold

Because of συχνά, the sentence clearly describes something that happens repeatedly, not just one time.

Yes. Greek allows quite a lot of flexibility in word order.

For example, these are all possible:

• Τον Δεκέμβριο μένω συχνά σπίτι το βράδυ...
• Το βράδυ τον Δεκέμβριο μένω συχνά σπίτι...
• Μένω συχνά σπίτι το βράδυ τον Δεκέμβριο...

The main meaning stays similar, but the first version is the most natural and balanced for a neutral statement.
The element placed earlier often gets a bit more emphasis.