Die Luft ist angenehm kühl.

Because Luft is a feminine noun in German, so its nominative singular article is die.

• die Luft = the air

This is something you usually just have to learn together with the noun:

• die Luft
• die Sonne
• die Zeit

In this sentence, die Luft is the subject, so die is the correct form.

German main clauses usually follow the verb-second rule. That means the finite verb comes in position 2.

In Die Luft ist angenehm kühl:

1. Die Luft
2. ist
3. angenehm kühl

This is very normal German word order for a simple statement.

Because here they are not before a noun. They come after the verb ist and describe the subject, so they are being used predicatively.

Compare:

• die angenehm kühle Luft = the pleasantly cool air

  • here the adjectives come before the noun, so they take endings

• Die Luft ist angenehm kühl = The air is pleasantly cool

  • here they come after ist, so no adjective endings

This is similar to English:

• the cool air
• the air is cool

You can think of angenehm here as modifying kühl, so in English it works a lot like pleasantly.

• kühl = cool
• angenehm kühl = pleasantly cool / nicely cool

German does not usually change the form between adjective and adverb the way English often does. So the same form can do both jobs.

That is why German says:

• angenehm kühl not
• angenehmly kühl or anything similar

German often stacks descriptive words like this, just like English can.

Here:

• kühl gives the basic quality: cool
• angenehm adds the speaker’s opinion: pleasant / pleasantly

So the phrase means something like:

• pleasantly cool
• nicely cool

It suggests the air is cool in a comfortable, enjoyable way, not too cold.

Yes, you could, but it is not exactly the same nuance.

• Die Luft ist angenehm kühl means the air is pleasantly cool.
  Angenehm modifies kühl.

• Die Luft ist kühl und angenehm means the air is cool and pleasant.
  Here the two qualities are presented more separately.

So both are correct, but angenehm kühl sounds a bit more natural if you want one combined idea.

Yes. You can say:

• die angenehm kühle Luft

That means:

• the pleasantly cool air

Notice what changes:

• after ist: angenehm kühl
• before a noun: angenehm kühle Luft

The adjective kühl gets the ending -e because it is now directly describing the noun Luft.
Angenehm stays unchanged because it is modifying kühle, not Luft directly.

Kühl contains ü, which can be difficult for English speakers.

A rough guide:

• k = like English k
• ü = a front rounded vowel; say something like ee with rounded lips
• hl = the l is pronounced, but the word is still quite short

A rough approximation is something like kyool, but that is only approximate. The real German ü does not exist in standard English.

Also note:

• kühl = cool
• schwül = muggy, humid

These are easy to mix up if pronunciation is unclear.

Because all nouns are capitalized in German.

So:

• die Luft
• das Haus
• der Tag

But adjectives and verbs are normally not capitalized:

• kühl
• angenehm
• ist

This is one of the most noticeable spelling differences from English.

Mostly yes, but with a nuance. Kühl refers to the air being cool, and angenehm adds that this coolness feels good or comfortable.

So it is not just a neutral weather description. It also expresses a positive impression:

• the air feels refreshingly cool
• the coolness is pleasant, not unpleasant

That emotional or sensory nuance is an important part of the sentence.