Söngkonan í hljómsveitinni syngur skýrt og talar líka fallega við áhorfendur.

Because Icelandic usually adds the definite article to the end of the noun instead of using a separate word like the.

• söngkona = female singer / singer
• söngkonan = the singer

Here, -an is the definite ending for a feminine noun in the nominative singular.

It modifies Söngkonan.

So the structure is:

• Söngkonan í hljómsveitinni = the singer in the band

It is describing which singer we mean. It is not mainly saying where the action happens, but identifying the singer.

Because í changes case depending on meaning.

When í means in in the sense of location, it usually takes the dative case. So:

• nominative definite: hljómsveitin = the band
• dative definite: hljómsveitinni = in the band

That is why the sentence has í hljómsveitinni.

They are both present tense, third person singular forms, because the subject is singular: Söngkonan.

• syngja = to sing → syngur = sings
• tala = to speak / to talk → talar = speaks / talks

So both verbs agree with the singular subject.

Because they are adverbs, not adjectives.

They describe how she sings and speaks:

• syngur skýrt = sings clearly
• talar fallega = speaks beautifully

Adverbs do not agree with the noun in gender or number the way adjectives do.

A useful thing to notice:

• skýr is an adjective meaning clear
• skýrt can be used adverbially to mean clearly
• fallegur / falleg / fallegt means beautiful
• fallega is the adverb beautifully

Líka means also. In Icelandic, it often comes after the finite verb or near the part of the sentence it emphasizes.

So:

• talar líka fallega = also speaks beautifully

This placement is very natural. English and Icelandic do not always place also in exactly the same spot, so this is something learners often need to get used to.

Because tala við is the normal Icelandic expression for talk to / speak to someone.

So:

• tala við einhvern = to speak to someone

Even though við can have other meanings in other contexts, here it is part of the verb phrase tala við.

Icelandic often uses a plural noun where English might use a collective singular.

• áhorfendur literally means spectators / audience members
• In natural English, that may be translated as the audience

So Icelandic is focusing on the people watching, as a group of individuals.

Because the noun here is indefinite plural.

• áhorfendur = spectators / audience members
• áhorfendurna = the spectators / the audience members in the accusative plural

So the sentence is literally closer to speaks beautifully to spectators / audience members, though English may translate that more naturally as to the audience.

It is in the accusative plural, because við takes the accusative in this expression.

The dictionary form is singular áhorfandi = spectator / viewer. Its plural is áhorfendur.

In this sentence:

• við áhorfendur = to the audience / to the spectators

Yes, in a way. Icelandic main clauses normally follow the V2 pattern, meaning the finite verb tends to come in the second position.

Here the first slot is the whole subject phrase:

• Söngkonan í hljómsveitinni = first slot
• syngur = finite verb in second position

So the sentence is perfectly normal Icelandic word order.

Not really two separate full clauses with repeated subjects. It joins two verbs that share the same subject.

The subject Söngkonan í hljómsveitinni applies to both:

• syngur skýrt
• talar líka fallega við áhorfendur

So Icelandic, like English, does not need to repeat the subject the second time.