Himlen er grå i dag.

Because himlen is the definite singular form of himmel, meaning sky.

• himmel = sky
• himlen = the sky

In Danish, the definite article is usually added to the end of the noun as a suffix.
So instead of a separate word like the, Danish often uses -en, -et, or -ne.

Also, with nouns like himmel, the spelling changes a little when the ending is added, so himmel + en becomes himlen, not himmelen in normal standard usage.

Because Danish usually puts the onto the noun itself.

So:

• en himmel = a sky
• himlen = the sky

This is a very common feature of Danish nouns. English uses a separate article before the noun, but Danish often uses a suffix instead.

Er is the present tense of at være, which means to be.

So:

• jeg er = I am
• du er = you are
• han er = he is
• vi er = we are

Unlike English, Danish does not change the present form depending on the subject. It stays er for everyone.

Because grå agrees with himlen, which is a singular common-gender noun.

After er, the adjective is in predicative position, and it still matches the noun:

• common gender singular: grå
• neuter singular: gråt

So:

• Himlen er grå = The sky is gray
• Huset er gråt = The house is gray

Since himmel is a common-gender noun, grå is the correct form here.

Because the standard Danish expression for today is i dag, written as two words.

It is a fixed adverbial phrase:

• i = in/on
• dag = day

But together i dag simply means today.

This is the normal modern spelling.

Because that is a very natural place for a time expression in Danish.

The sentence structure here is:

• Himlen = subject
• er = verb
• grå = complement
• i dag = time expression

So the pattern is basically:

Subject + verb + description + time

That said, Danish is flexible, and you can also say:

I dag er himlen grå.

That version is also correct, just with more focus on today.

Yes, absolutely.

Both are correct:

• Himlen er grå i dag.
• I dag er himlen grå.

The difference is mainly emphasis.

Danish main clauses follow the verb-second rule. That means the finite verb usually stays in the second position:

• Himlen / er / grå i dag
• I dag / er / himlen grå

So when i dag moves to the front, er still has to come second.

No. In Danish, nouns are not capitalized just because they are nouns.

Himlen has a capital H only because it is the first word of the sentence.

So in the middle of a sentence, it would normally be written himlen, not Himlen.

You would say den grå himmel.

This is useful because Danish treats adjectives differently depending on where they appear:

• Himlen er grå = The sky is gray
  
  • adjective after the verb
• den grå himmel = the gray sky
  
  • adjective directly before the noun

So when the adjective comes before a definite noun, Danish usually uses den/det/de plus the adjective and noun.

The letter å is pronounced roughly like the vowel in English law or saw, though the exact Danish sound is a bit different.

So grå is roughly like graw.

A few pronunciation notes:

• å is an important Danish vowel
• it does not sound like a in cat
• the r in Danish is also different from English, so the word may sound harder than it looks

If you are learning pronunciation, grå is a good word to practice because it contains the very common Danish letter å.