Símsvarinn tekur við skilaboðunum núna.

Because -inn is the suffixed definite article in Icelandic.
So:

• símsvari = answering machine
• símsvarinn = the answering machine

In this sentence, símsvarinn is the subject, so it is in the nominative singular. Icelandic usually attaches the to the end of the noun instead of using a separate word like English does.

Because taka við is a fixed verb expression meaning to receive, to accept, or to take in.

So here:

• taka by itself = take
• taka við = receive / accept

That means Símsvarinn tekur við skilaboðunum is not literally just The answering machine takes the messages, but more idiomatically The answering machine receives the messages.

This kind of verb + particle combination is very common in Icelandic.

Because taka við takes the dative case, and skilaboðunum is the dative plural definite form.

Here is the pattern:

• skilaboð = messages
• skilaboðin = the messages (nominative/accusative)
• skilaboðunum = to/for the messages, or in this case the form required after taka við

So the sentence uses skilaboðunum because the verb expression demands it.

A useful thing to memorize is:

• taka við einhverju = receive something

where einhverju shows that the noun after it goes into the dative.

Grammatically, tekur is the 3rd person singular present tense of taka.

In this sentence, because of the full expression tekur við, the meaning is receives or is receiving.

Icelandic does not have a separate progressive form like English is receiving in the same way English does. The simple present often covers both:

• receives
• is receiving

Here, because of núna (now), English will often translate it more naturally as is receiving now or is taking messages now.

Núna means now. It tells you that this is happening at the present moment.

Without núna, the sentence could sound more general:

• Símsvarinn tekur við skilaboðunum.
  = The answering machine receives/is receiving the messages.

With núna, it becomes more immediate:

• Símsvarinn tekur við skilaboðunum núna.
  = The answering machine is receiving the messages now.

So núna is not required for the grammar, but it adds time reference and emphasis.

Yes, somewhat. Icelandic has fairly flexible word order, but main clauses usually follow the verb-second rule. That means the finite verb often stays in the second position.

The sentence as given is very natural:

• Símsvarinn tekur við skilaboðunum núna.

You can also front núna for emphasis:

• Núna tekur símsvarinn við skilaboðunum.

That still works because tekur remains in second position.

So word order can change for emphasis, but not completely freely.

A rough English-friendly pronunciation would be:

SEEMS-sva-rin TEH-kur vith SKIH-la-bo-thu-num NOO-na

A few helpful notes:

• Sí- sounds like see
• sv in símsvarinn is pronounced clearly
• ð is like the th in this, not the th in thin
• ú sounds like oo
• stress is usually on the first syllable in Icelandic words

So the rhythm is roughly:

SÍM-svarinn TE-kur við SKI-la-bo-ðu-num NÚ-na