Snjórinn bráðnar þegar sólin skín.

They are each a noun plus the suffixed definite article:

• snjórinn = snjór
  • -inn = the snow
• sólin = sól
  • -in = the sun

In Icelandic, the is usually attached to the end of the noun instead of appearing as a separate word.

Also, the article changes form depending on the noun’s gender, case, and number, so -inn and -in are not interchangeable endings.

Because Icelandic normally uses a suffixed article rather than a separate article like English the.

So instead of saying something like the snow, Icelandic usually says:

• snjór = snow
• snjórinn = the snow

And:

• sól = sun
• sólin = the sun

A separate word can be used in some special cases, but for ordinary sentences like this one, the attached article is the normal pattern.

They are both in the nominative singular.

That is because each one is the subject of its clause:

• Snjórinn bráðnar = the snow melts
• þegar sólin skín = when the sun shines

Since both the snow and the sun are doing the action in their own clauses, nominative is the expected case.

Because Icelandic verbs do not all form the present tense in exactly the same way.

Here you have two different verbs:

• bráðna = to melt
  
  • bráðnar = melts / is melting
• skína = to shine
  
  • skín = shines / is shining

Both are 3rd person singular present, but they belong to different verb patterns, so their present forms look different.

This is very normal in Icelandic: you often have to learn a verb’s principal forms rather than assuming one universal ending.

It can cover both, depending on context.

Icelandic often uses the simple present where English might use either:

• simple present: melts, shines
• present progressive: is melting, is shining

In a sentence like this, it most naturally sounds like a general truth or regular occurrence, so English usually translates it with the simple present:

• The snow melts when the sun shines.

But the Icelandic present itself is not as narrowly restricted as English simple present.

Here, þegar means when and introduces a time clause:

• þegar sólin skín = when the sun shines

A useful extra point: þegar can also mean already in other contexts. For example:

• Ég er þegar kominn = I have already arrived

So learners often need to tell from context whether þegar means when or already. In your sentence, it is clearly the conjunction when.

Yes. The sentence can also be written with the þegar-clause first:

• Þegar sólin skín, bráðnar snjórinn.

That still means the same thing.

The important thing to notice is that Icelandic follows a verb-second pattern in main clauses. So when the subordinate clause comes first, the finite verb in the main clause still comes early:

• Þegar sólin skín, bráðnar snjórinn
  • not Þegar sólin skín, snjórinn bráðnar

That word order is very important in Icelandic.

Because bráðna is intransitive: it means to melt in the sense of becoming melted.

• Snjórinn bráðnar = The snow melts

By contrast, bræða is transitive: it means to melt something.

• Sólin bræðir snjóinn = The sun melts the snow

So in your sentence, the snow itself is undergoing the change, which is why bráðna is the right verb.

The dictionary forms are:

• snjórinn → snjór
• bráðnar → bráðna
• þegar → þegar
• sólin → sól
• skín → skína

For nouns, dictionaries usually list the form without the suffixed article.
For verbs, dictionaries usually list the infinitive.

So if you want to look these words up, those are the forms you would normally search for.

A rough learner-friendly guide:

• ó in snjórinn sounds roughly like o in go, often with a slight glide
• á in bráðnar sounds roughly like ow in now
• í in skín sounds like ee in see
• ð in bráðnar is like th in this, though in real speech it can sound very weak, especially before n

Also, Icelandic stress usually falls on the first syllable of the word, so:

• SNJÓ-rinn
• BRÁÐ-nar
• SÓ-lin
• SKÍN

That first-syllable stress pattern is very consistent in Icelandic and helps a lot with pronunciation.