Ich trinke den warmen Kakao am Abend.

Because Kakao is the direct object of the verb trinken.

In German, the direct object usually takes the accusative case.
The noun Kakao is masculine, so:

• der warme Kakao = nominative (used for the subject)
• den warmen Kakao = accusative (used for the direct object)

In this sentence:

• Ich = the subject
• trinke = the verb
• den warmen Kakao = the thing being drunk, so it must be accusative

That is why both the article and the adjective change.

Because adjectives in German change their endings depending on:

• the case
• the gender
• whether there is an article before the noun

Here, Kakao is:

• masculine
• accusative
• preceded by the definite article den

So the adjective takes the ending -en:

• den warmen Kakao

This is a very common pattern. For example:

• der kalte Tee → nominative masculine
• den kalten Tee → accusative masculine

Am Abend is a common German way to say in the evening or in the evenings, depending on context.

It gives a time expression telling you when the action happens.

So in this sentence, am Abend tells us the drinking happens in the evening.

German often uses:

• am Morgen = in the morning
• am Nachmittag = in the afternoon
• am Abend = in the evening

Am is a contraction of:

• an dem → am

So am Abend literally comes from an dem Abend.

You do not usually say an dem Abend in a normal sentence like this unless you mean a very specific evening for emphasis.
In everyday German, am Abend is the natural form.

Because German main clauses usually follow the verb-second rule.

In a normal statement, the finite verb comes in the second position:

• Ich trinke den warmen Kakao am Abend.

Here:

1. Ich
2. trinke

That is standard German word order.

If you move something else to the front, the verb still stays second:

• Am Abend trinke ich den warmen Kakao.

So the verb trinke remains in second position either way.

Yes, absolutely.

Both are correct:

• Ich trinke den warmen Kakao am Abend.
• Am Abend trinke ich den warmen Kakao.

The difference is mainly emphasis:

• Ich trinke ... am Abend = neutral statement
• Am Abend trinke ich ... = puts more focus on when

Notice that when am Abend moves to the front, the verb still has to stay second, so it becomes:

• Am Abend trinke ich ... not
• Am Abend ich trinke ...

Because they are nouns, and all nouns are capitalized in German.

So in this sentence:

• Kakao is a noun
• Abend is a noun

This is one of the most noticeable spelling rules in German.
Even abstract nouns and time nouns are capitalized:

• das Haus
• die Zeit
• der Abend

It can mean either one, depending on context.

German does not usually have a separate everyday form like English I am drinking.
The present tense ich trinke can mean:

• I drink
• I am drinking

In this sentence, because of am Abend, it often sounds like a habitual action:

• I drink warm cocoa in the evening

But depending on the situation, it could also describe what is happening now.

Both can refer to the evening, but they are used a little differently.

• am Abend = in the evening / on the evening
• abends = in the evenings / evenings, often with a habitual sense

Examples:

• Ich trinke den warmen Kakao am Abend.
  This can mean in the evening in a general or specific context.

• Ich trinke abends warmen Kakao.
  This often sounds more like a routine: I drink warm cocoa in the evenings.

So abends is often a good choice when you want to emphasize repeated habit.

There actually is an article there.

Am = an dem

So the article is dem, just hidden inside the contraction.

This is very common in German with prepositions and articles:

• an dem → am
• in dem → im
• zu dem → zum
• zu der → zur

So am Abend already includes the article.

Yes, you can, but it changes the feel slightly.

• den warmen Kakao = a more specific cocoa, like the warm cocoa
• warmen Kakao = warm cocoa in a more general sense

So both are grammatical:

• Ich trinke den warmen Kakao am Abend.
• Ich trinke warmen Kakao am Abend.

The version with den sounds more definite and specific.
The version without an article sounds more like you are talking about the drink as a substance in general.