Skulum við byrja á súpu?

Skulum is the we form of the verb skulu. In sentences like this, it is often used to make a suggestion, so it works a lot like English shall we or let’s.

So:

• Við skulum byrja ... = Let’s start ...
• Skulum við byrja ...? = Shall we start ...?

It is not a direct word-for-word match with English, but in this sentence it signals a polite proposal.

Because this is a yes/no question.

In Icelandic, yes/no questions usually put the finite verb first:

• Við skulum byrja á súpu. = Let’s start with soup.
• Skulum við byrja á súpu? = Shall we start with soup?

So the subject við comes after the verb in the question.

Byrja means to begin / to start, and byrja á means to start with something.

So:

• byrja á súpu = start with soup

This is a common Icelandic pattern: a verb plus a preposition that does not always translate literally. Here, á is just the preposition that goes with byrja when you mean start with something.

Because á in this expression takes the dative case, and súpa changes form in the dative singular.

The noun is:

• súpa = nominative singular
• súpu = dative singular

Since the phrase is byrja á + dative, you get:

• á súpu

This is a very common thing in Icelandic: prepositions often control a particular case, and the noun changes form accordingly.

Yes, in this sentence it is.

Skulum tells you that the subject is we, but Icelandic normally still includes the pronoun við in a full sentence like this:

• Skulum við byrja á súpu?

Leaving it out would sound incomplete or unusual in ordinary speech.

Yes, it sounds natural and polite.

It could be used when looking at a menu or deciding what to eat first:

• Skulum við byrja á súpu? = Shall we start with soup?

It has a slightly polite, cooperative tone, like inviting the other person to agree.

Yes. A very common alternative is:

• Eigum við að byrja á súpu?

This also means Shall we start with soup?

Very roughly:

• Skulum við ...? can sound like a suggestion with shall we / let’s
• Eigum við að ...? is also a very common way to ask shall we / should we ...?

Both are natural, but Skulum við byrja á súpu? is perfectly good Icelandic.

Here are the main forms:

• skulum → from skulu = a modal verb often meaning shall / will / be supposed to
• við = we
• byrja = to begin / start
• á = a preposition; here it is part of byrja á = start with
• súpu → from súpa = soup

So even though the full meaning is not built by translating each word one by one, knowing the dictionary forms helps you recognize the sentence structure.