Hún vaknaði úr dáinu.

In Icelandic, vakna already means wake up / awaken. So Hún vaknaði naturally means she woke up.

English often uses a particle like up, but Icelandic does not need one here. You can sometimes see vakna upp, but plain vakna is the normal basic verb.

Vaknaði is the 3rd person singular past tense of vakna.

So:

• vakna = to wake up
• vaknaði = woke up

It matches hún (she), which is third person singular.

This is a very common and useful distinction:

• vakna = to wake up, to become awake
  
  • intransitive: the subject wakes
• vekja = to wake someone, to awaken someone
  
  • transitive: someone causes another person to wake

So in this sentence, hún is the one who wakes up herself, so vakna is the correct verb.

Compare:

• Hún vaknaði. = She woke up.
• Ég vakti hana. = I woke her up.

Úr usually means out of or from, and it is often used when someone or something emerges from a state, condition, or enclosed situation.

So vakna úr dái / úr dáinu is the normal way to say wake from a coma.

The idea is literally something like come out of the coma-state.

Because the preposition úr takes the dative.

The dictionary form is dá. After úr, it changes to the dative form:

• dá = coma
• úr dái = out of a coma
• úr dáinu = out of the coma

So the case is not chosen by the verb here; it is required by the preposition úr.

Because dáinu is the definite form: the coma.

Compare:

• úr dái = out of a coma / out of coma
• úr dáinu = out of the coma

The ending -nu is part of the suffixed definite article in this dative singular form. Icelandic usually puts the at the end of the noun rather than as a separate word.

Normally, you do need it.

Icelandic is not a language that usually drops subject pronouns in ordinary sentences. So:

• Hún vaknaði úr dáinu. is the normal full sentence.

Just saying Vaknaði úr dáinu would usually sound incomplete unless the subject is understood from a very specific context.

Yes, but the basic order here is the most neutral one:

• Hún vaknaði úr dáinu.
  subject + verb + prepositional phrase

Icelandic word order is fairly flexible, but it follows a strong verb-second pattern. That means if you move another element to the front, the finite verb still stays in second position:

• Úr dáinu vaknaði hún.

That version is possible, but it sounds more marked or stylistic. The original sentence is the most straightforward way to say it.