Við fengum sitthvora fötuna og byrjuðum að þrífa svalirnar.

Sitthvora means something like one each or a separate one for each person.

In this sentence, Við fengum sitthvora fötuna means that each person got their own bucket. So it is not one shared bucket for everyone, but one bucket per person.

A very natural English translation would be:

• We each got a bucket
• We got a bucket each

Because it has to agree with fötuna in gender, number, and case.

Here, fötuna is:

• feminine
• singular
• accusative

So sitthvora also appears in the matching form: sitthvora.

This word changes form depending on the noun it goes with, much like an adjective or determiner.

Because Icelandic often uses the singular in distributive expressions like this, where each person gets one item.

So:

• sitthvora fötuna = a bucket each
• literally, something like their respective bucket

Even though several buckets are involved in the real-world meaning, the grammar stays singular because each individual person is associated with one bucket.

Fötuna is the definite form: the bucket.

In expressions with sitthvor, Icelandic commonly uses the definite noun, even though English usually uses an indefinite expression:

• sitthvora fötuna
  literally: the separate bucket for each
• natural English: a bucket each

So this is one of those places where Icelandic and English structure things differently.

It is in the accusative case because it is the direct object of fengum (got/received).

The verb fá usually takes a direct object in the accusative.

So in:

• Við fengum sitthvora fötuna

the thing received is sitthvora fötuna, so it appears in the accusative.

Fengum is the 1st person plural past tense of fá (to get, receive).

So:

• ég fékk = I got
• við fengum = we got

The verb fá is irregular, so the past tense does not look very similar to the infinitive.

It means started cleaning.

The verb byrja often appears with:

• byrja að + infinitive

So:

• byrjuðum = we started
• að þrífa = to clean

Together:

• byrjuðum að þrífa = we started to clean / we started cleaning

This is a very common Icelandic pattern.

Because after byrja (to begin/start), Icelandic normally uses að + infinitive.

So:

• byrja að lesa = start reading
• byrja að vinna = start working
• byrja að þrífa = start cleaning

This að is the ordinary infinitive marker, similar to English to.

Because svalir is normally a plural noun in Icelandic.

Even when English would often say the balcony in the singular, Icelandic commonly says svalir.

So:

• svalir = balcony / balconies
• svalirnar = the balcony / the balconies

In many contexts, svalirnar may be best translated as the balcony, even though the Icelandic form is plural.

It breaks down like this:

• svalir = balcony/balconies
• -nar = the definite article ending for this plural form

So svalirnar means the balcony or the balconies, depending on context and translation style.

No, this is a fairly normal Icelandic main-clause word order:

• Við = subject
• fengum = verb
• sitthvora fötuna = object
• og byrjuðum að þrífa svalirnar = and started cleaning the balcony

So the structure is basically:

• We got a bucket each and started cleaning the balcony

Nothing especially unusual is happening with word order here; the main challenges are the forms sitthvora, fötuna, and svalirnar.