Við göngum heim saman eftir kvöldmat.

Yes:

• Við = we
• göngum = walk / are walking (1st person plural present of að ganga)
• heim = home, more literally homeward
• saman = together
• eftir = after
• kvöldmat = dinner / the evening meal (from kvöldmatur)

So the structure is basically:

We walk home together after dinner.

Yes. við can be:

• a pronoun meaning we
• a preposition with meanings like with, by, or at, depending on context

Here it means we because it is the subject of the sentence and comes right before the verb göngum.

So in this sentence:

• Við göngum ... = We walk ...

not With walk ...

Because Icelandic verbs change form depending on the subject.

The basic verb is að ganga = to walk.
But with við = we, you need the 1st person plural present form:

• við göngum = we walk

So ganga is the dictionary form / infinitive, while göngum is the form that matches we.

The vowel change is just part of this verb’s pattern, so this is something you mostly learn with the verb itself.

In this sentence, heim works as an adverb meaning home or homeward.

Icelandic often says:

• fara heim = go home
• koma heim = come home
• ganga heim = walk home

So you do not need a separate word meaning to before heim.
The idea of movement toward home is already built into heim here.

saman means together.

It tells you that the people are walking home as a group. If you remove it, the sentence still works:

• Við göngum heim eftir kvöldmat. = We walk home after dinner.

That version just doesn’t explicitly say that they do it together.

So saman is optional for grammar, but important for meaning.

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

So kvöldmat is functioning as dative singular here.

A detail that often confuses learners: with kvöldmatur, the accusative singular and dative singular look the same:

• accusative: kvöldmat
• dative: kvöldmat

So even though the form does not change visibly, the case is still dative after eftir in this sentence.

Because Icelandic, like English, often uses meal words without an article when speaking generally.

So:

• eftir kvöldmat = after dinner

This is the normal general expression.

If you said eftir kvöldmatinn, that would mean something more like:

• after the dinner
• after that specific dinner

So the version without the article is the natural one for a general routine or event.

Yes, word order with adverbs is somewhat flexible in Icelandic.

The given sentence:

• Við göngum heim saman eftir kvöldmat.

is natural, and it groups the ideas as:

• walk home together
• after dinner

But you can also hear:

• Við göngum saman heim eftir kvöldmat.

That is also natural and means essentially the same thing.

So heim saman is not the only possible order. The sentence you were given is a normal Icelandic way to say it, but adverb placement can vary.

It can mean either, depending on context.

Icelandic present tense often covers both:

• a habitual meaning: We walk home together after dinner
• a current meaning: We’re walking home together after dinner

Usually the situation makes it clear.

If you want to make right now more obvious, Icelandic can add context words such as núna (now) or use other constructions, but the simple present by itself is very common.

The dictionary form is kvöldmatur.

That is a masculine noun meaning dinner / evening meal.

In the sentence, you see kvöldmat instead of kvöldmatur because the noun is not in the nominative dictionary form. After eftir, it appears in the required case form, which here is dative singular, though it happens to look the same as the accusative singular.