Solen er varm.

Because solen is the definite singular form of sol.

• sol = sun / a sun
• solen = the sun

In Norwegian, the definite article is usually added to the end of the noun as a suffix, instead of being a separate word like English the.

In Norwegian, the is often built into the noun itself.

So instead of:

• English: the sun
• Norwegian: solen

This is very common in Scandinavian languages. The ending -en here marks the noun as definite.

A separate determiner can appear in some other structures, for example:

• den varme solen = the warm sun

But in Solen er varm, the definiteness is already shown by the ending -en.

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

So:

• jeg er = I am
• du er = you are
• han/hun er = he/she is
• solen er = the sun is

Unlike English, Norwegian verbs do not change depending on the subject. So er works for all persons in the present tense.

Because the adjective must agree with the noun, and here it is describing a singular common-gender noun.

Solen is treated here as a common-gender singular noun, so the adjective is:

• varm = common gender singular

Compare:

• en varm dag = a warm day
• et varmt hus = a warm house
• varme dager = warm days

Since solen is singular and not neuter, varm is the correct form.

Because varm is being used after the verb er. This is called a predicative adjective, and in Norwegian predicative adjectives do not take the same definite form that adjectives before nouns do.

Compare:

• den varme solen = the warm sun
  
  • adjective is before the noun, so it becomes varme
• Solen er varm = The sun is warm
  
  • adjective is after er, so it stays in the predicative form varm

So this difference is about position in the sentence, not just about definiteness.

This is a good question, because sol can be a bit flexible in Bokmål.

Traditionally, sol is a feminine noun, so many learners also see:

• sola = the sun

But in Bokmål, many feminine nouns can also be treated as common/masculine in practice, giving:

• solen

So both sola and solen can be correct in Bokmål, depending on style and preference.
In this sentence, solen is perfectly acceptable.

Yes, in this simple sentence it is basically the same:

• Solen = subject
• er = verb
• varm = adjective/complement

So the pattern is:

Subject + verb + adjective

That said, Norwegian is a V2 language, which means that in many main clauses the verb must come in the second position. In this very simple sentence, that rule gives the same order English uses.

A rough pronunciation guide is:

• Solen ≈ SOO-len
• er ≈ air or a lighter ehr, depending on accent
• varm ≈ varm with a rolled or tapped r in many accents

A rough IPA version in one common pronunciation is:

• Solen er varm ≈ /ˈsuːlən eːr varm/

A few helpful notes:

• o in solen is a long oo sound
• the r in Norwegian is usually clearer than in many English accents
• final consonants are usually pronounced clearly

Exact pronunciation varies by region, but that approximation is good for a learner.

Because it is the first word of the sentence.

Norwegian capitalizes:

• the first word of a sentence
• proper names

It does not capitalize ordinary nouns the way German does.

So:

• Solen er varm. = correct
• solen er varm. = only correct if it is not starting a sentence
• Solen Er Varm. = incorrect