Eftir rigninguna kemur fallegur regnbogi yfir húsunum.

This is because Icelandic follows a verb-second pattern in main clauses.

When the sentence starts with an adverbial phrase like Eftir rigninguna meaning after the rain, the finite verb usually comes next:

• Eftir rigninguna = first element
• kemur = second element
• fallegur regnbogi = subject

So the structure is:

• Eftir rigninguna kemur fallegur regnbogi yfir húsunum.

A more basic word order would be:

• Fallegur regnbogi kemur yfir húsunum eftir rigninguna.

That is also grammatical, but it puts less emphasis on after the rain.

Because eftir here takes the accusative case, and rigninguna is the definite accusative singular form of rigning meaning rain.

Here is the breakdown:

• rigning = rain
• rigninguna = the rain

Why not rigningin?

• rigningin is nominative definite singular
• but after eftir in the sense of after, Icelandic uses the accusative

So:

• eftir rigninguna = after the rain

Not always. Icelandic prepositions can take different cases depending on meaning.

With eftir, the most common meanings are:

• accusative for after
• dative in some other uses, such as according to or certain expressions involving movement along something

In this sentence, eftir clearly means after, so the accusative is used:

• eftir rigninguna

Because fallegur agrees with regnbogi in:

• gender
• number
• case

Regnbogi is:

• masculine
• singular
• nominative

So the adjective must also be:

• masculine
• singular
• nominative

That gives:

• fallegur regnbogi = a beautiful rainbow

If the noun changed, the adjective would change too.

Because regnbogi is the subject of the verb kemur.

Even though the subject comes after the verb here, it is still the subject of the sentence:

• kemur fallegur regnbogi = a beautiful rainbow comes / appears

Subjects are normally in the nominative in Icelandic.

Kemur is the 3rd person singular present tense of koma meaning to come.

It is singular because the subject is singular:

• fallegur regnbogi = a beautiful rainbow

So:

• ég kem = I come
• þú kemur = you come
• hann / hún / það kemur = he / she / it comes

Here the rainbow is treated as a singular subject, so kemur is the correct form.

Icelandic does not have an indefinite article like English a or an.

So:

• fallegur regnbogi can mean a beautiful rainbow
• context tells you it is indefinite

For the definite form, Icelandic usually adds the article to the end of the noun:

• regnboginn = the rainbow

So the contrast is:

• fallegur regnbogi = a beautiful rainbow
• fallegi regnboginn = the beautiful rainbow

Húsunum is the dative plural definite form of hús meaning houses.

Breakdown:

• hús = house / houses
• húsunum = the houses

It is in the dative because of the preposition yfir in this sentence.

Here yfir húsunum means over the houses in a location or position sense, not movement toward/onto them. That is why the dative is used.

It can take either, depending on the meaning.

A helpful rule is:

• accusative if there is movement toward/onto/across something
• dative if it describes position or location

In this sentence:

• yfir húsunum = over the houses

The rainbow is being described as appearing above the houses, so this is a location idea, and the dative is natural.

A close gloss is:

• Eftir = after
• rigninguna = the rain
• kemur = comes
• fallegur = beautiful
• regnbogi = rainbow
• yfir = over
• húsunum = the houses

So the structure is roughly:

• After the rain comes a beautiful rainbow over the houses

That sounds a bit poetic in English, but it is perfectly natural in Icelandic.

Yes. You could say:

• Fallegur regnbogi kemur yfir húsunum eftir rigninguna.

That is grammatical, but the emphasis changes.

Compare:

• Eftir rigninguna kemur fallegur regnbogi yfir húsunum.
  Focuses first on after the rain

• Fallegur regnbogi kemur yfir húsunum eftir rigninguna.
  Focuses first on a beautiful rainbow

Icelandic often moves elements to the front for emphasis, while still keeping the finite verb in second position.

Grammatically, yes: koma means to come.

But in English, depending on context, you might translate the whole sentence more naturally as:

• After the rain, a beautiful rainbow appears over the houses.
• After the rain, a beautiful rainbow comes out over the houses.

So kemur is literally comes, but the best English translation may use appears.

Yes, regnbogi is masculine, and yes, you do need to learn noun gender in Icelandic.

Gender matters because it affects:

• adjective endings
• articles
• pronouns
• sometimes other parts of the sentence

Here, because regnbogi is masculine, you get:

• fallegur regnbogi

If it were a feminine noun, the adjective ending would be different. So learning nouns together with their gender is very important in Icelandic.