Eftir viðvörun í símanum var okkur sýnt skýlið við hliðina á húsinu, og þar biðum við rólega.

Here eftir means after in a time sense, and in that use it takes the accusative.

The noun viðvörun is being used indefinitely here: after a warning, not after the warning. If the speaker meant one specific, already identified warning, you could have eftir viðvörunina.

A small extra point: viðvörun looks the same in nominative and accusative singular, so the case is not visible from the form alone.

Literally, yes, í usually means in. But in natural English, viðvörun í símanum is best understood as something like a warning on the phone or an alert on the phone.

Grammatically:

• í takes the dative when it indicates location.
• símanum is the dative singular definite form of sími (phone).

So í símanum = in/on the phone.

Because okkur is the dative/object form of við.

• við = we (subject form, nominative)
• okkur = us / to us (object form, here dative)

The verb sýna normally works like this:

• sýna einhverjum eitthvað = show someone something

So in var okkur sýnt skýlið, okkur means to us.

This is a passive-style way of saying that someone showed the shelter to us.

A more active version would be:

• Einhver sýndi okkur skýlið = Someone showed us the shelter

In the sentence you gave:

• var = was
• okkur = to us
• sýnt = shown
• skýlið = the shelter

So the structure is basically was shown to us the shelter, even though English would normally say the shelter was shown to us or we were shown the shelter.

Icelandic often allows this kind of ordering quite naturally.

Because okkur is not the nominative subject here, so the verb does not agree with it.

The plural people are expressed by okkur (to us), which is dative. That does not make the verb plural.

Also, skýlið (the shelter) is singular, so a singular verb form is exactly what you would expect in a passive expression like this.

Sýnt is the past participle of sýna (to show) in the neuter singular form.

That form is used here because:

• skýlið is neuter singular
• and neuter singular is also a very common form in passive/impersonal constructions

So sýnt is the natural form here.

A useful comparison:

• Bókin var sýnd = The book was shown
• Húsið var sýnt = The house was shown

Since skýlið is neuter, sýnt fits it.

Skýlið is the definite singular form of skýli, meaning shelter.

• skýli = a shelter
• skýlið = the shelter

The ending -ið is the suffixed definite article for this neuter noun.

Also, for many neuter nouns in Icelandic, the nominative and accusative singular look the same, and skýlið is one of those forms.

This is a very common expression meaning next to / beside.

It is best learned as a whole chunk:

• við hliðina á = next to / beside

Literally, it is something like by the side of.

So:

• við hliðina á húsinu = beside the house

Learners often find this phrase confusing because it contains more than one preposition-like element, but in practice it is easiest just to memorize við hliðina á as a fixed expression.

Because in the expression við hliðina á, the noun after á is normally in the dative.

So:

• húsinu = dative singular of húsið / hús
• á húsinu here is part of the fixed phrase við hliðina á húsinu

That is why you get húsinu, not húsið.

This is because Icelandic follows the V2 rule very often: the finite verb tends to come in the second position of the clause.

Here the clause begins with þar (there), so the verb comes next:

• þar = first element
• biðum = second element
• við = subject after the verb

So:

• þar biðum við rólega

is a very natural Icelandic word order.

You could also say við biðum þar rólega, but the given version puts a little more focus on there, linking it neatly to the shelter/location just mentioned.

Rólega is an adverb, meaning calmly or quietly.

It comes from the adjective rólegur (calm, quiet).

So:

• rólegur = calm/quiet
• rólega = calmly/quietly

It describes how they waited:

• biðum rólega = waited calmly

This is a very common pattern in Icelandic: an adjective can have a related adverb form used to describe the manner of an action.