Какой красивый закат!

Because какой here means what a / what an in an exclamation:

• Какой красивый закат! = What a beautiful sunset!

By contrast, как usually means how:

• Как красиво! = How beautiful!

So Russian uses какой + adjective + noun for this kind of What a ...! exclamation.

It has the same form as the question word какой (which? / what kind of?), but in this sentence it is being used exclamatorily, not as a real question.

Compare:

• Какой это закат? = What kind of sunset is this? → real question
• Какой красивый закат! = What a beautiful sunset! → exclamation

So the form is the same, but the function is different.

They both have to agree with закат.

Закат is:

• masculine
• singular
• nominative case

So the words describing it also appear in the same form:

• какой = masculine singular nominative
• красивый = masculine singular nominative
• закат = masculine singular nominative

Russian agreement is very important: adjectives and words like какой change their endings to match the noun.

Because this sentence is basically just an exclamatory noun phrase: What a beautiful sunset!

There is no verb here, and закат is simply the thing being identified or pointed out emotionally. In that kind of exclamation, Russian normally uses the nominative.

So:

• закат = nominative singular
• красивый matches it in nominative singular
• какой also matches it in nominative singular

No. Russian often leaves out the verb to be in the present tense.

English says:

• It is a beautiful sunset
• What a beautiful sunset it is!

Russian can simply say:

• Красивый закат
• Какой красивый закат!

That sounds completely natural. In emotional exclamations, Russian very often uses short verbless sentences like this.

The neutral, natural order for this exclamation is:

• Какой красивый закат!

This is the most straightforward way to say What a beautiful sunset!

Russian word order is flexible, but changing it usually changes the emphasis or makes it sound more literary, dramatic, or less natural in everyday speech.

For example:

• Какой закат красивый! is possible, but it sounds more marked and less standard for this meaning.
• Красивый закат! simply means Beautiful sunset!
• Закат какой красивый! is more expressive and conversational, something like That sunset is so beautiful!

So the original order is the best default pattern to learn.

Yes, but it is slightly different.

• Какой красивый закат! = What a beautiful sunset!
• Такой красивый закат! = Such a beautiful sunset! / So beautiful a sunset!

Both are exclamatory and natural, but какой is the closer match to English what a ...!

The stress is:

• како́й краси́вый зака́т

A rough pronunciation guide:

• како́й ≈ ka-KOY
• краси́вый ≈ kra-SEE-vy
• зака́т ≈ za-KAT

The intonation should sound clearly exclamatory, with emotion or admiration. The final exclamation mark is important for how you say it.

No. Even though какой can mean which in other contexts, that is not what it means here.

In this sentence, какой is part of an exclamatory pattern:

• Какой красивый закат! = What a beautiful sunset!

A literal word-for-word translation would be misleading. It is better to learn the whole pattern as one chunk.

No. Russian has no articles like a or the.

English needs:

• What a beautiful sunset!

Russian simply says:

• Какой красивый закат!

The meaning that English expresses with a is understood from context and from the exclamatory structure.

Yes, very often. This is a very useful pattern:

• Какой красивый дом! = What a beautiful house!
• Какая хорошая идея! = What a good idea!
• Какое тёплое море! = What a warm sea!
• Какие интересные люди! = What interesting people!

Notice how какой changes to match the noun:

• какой for masculine singular
• какая for feminine singular
• какое for neuter singular
• какие for plural

So in your sentence, какой is masculine because закат is masculine.