Við göngum yfir þessa brú saman.

Because ganga is the infinitive: að ganga = to walk.

In the sentence, the subject is við = we, so the verb has to be in the 1st person plural present form:

• ég geng = I walk
• þú gengur = you walk
• hann/hún/það gengur = he/she/it walks
• við göngum = we walk

So Við göngum... means We walk... / We are walking...

That vowel change is just part of how this verb is conjugated. Icelandic has many common verbs whose stem changes in some forms.

So with ganga, you should learn the forms as they are, rather than expecting the vowel to stay the same all the time:

• að ganga
• ég geng
• við göngum

This is normal Icelandic verb behavior, not something special about this sentence.

Here is the breakdown:

• Við = we
• göngum = walk / are walking
• yfir = across / over
• þessa = this (feminine singular accusative)
• brú = bridge
• saman = together

So the structure is:

• subject: Við
• verb: göngum
• prepositional phrase: yfir þessa brú
• adverb: saman

Because the preposition yfir takes the accusative here, and þessi has to match that case.

Brú is a feminine singular noun.
The demonstrative þessi changes by case:

• þessi brú = this bridge (nominative)
• þessa brú = this bridge (accusative)

In this sentence, the noun phrase is the object of yfir, so it appears in the accusative: þessa brú.

Because Icelandic prepositions often control different cases depending on meaning.

With yfir, a very common pattern is:

• accusative for movement across/over
• dative for location above/over

Here, the sentence describes movement: walking across the bridge.
So Icelandic uses accusative:

• yfir þessa brú

A location example would be something like above the table, where dative is more likely.

Because brú happens to have the same form in the nominative singular and accusative singular.

So:

• nominative: brú
• accusative: brú

That is why the case change is only visible in þessa, not in the noun itself.

Icelandic usually expresses the in one of two ways:

1. with a demonstrative like þessi = this
2. with a definite article attached to the noun, like brúin = the bridge

In this sentence, þessa brú already means this bridge, so there is no need for a separate word meaning the.

It can mean both, depending on context.

The Icelandic simple present often covers:

• habitual present: we walk
• current action: we are walking

So Við göngum yfir þessa brú saman could mean either one.

If you want to make the ongoing action especially clear, Icelandic often uses:

• Við erum að ganga yfir þessa brú saman
  = We are walking across this bridge together

Saman means together.

It is an adverb, and adverbs are often placed late in the clause in Icelandic, especially after the main object or prepositional phrase. So this word order sounds very natural:

• Við göngum yfir þessa brú saman.

Putting saman elsewhere is sometimes possible, but the end position is simple and neutral here.

Yes, but Icelandic has an important rule in main clauses: the finite verb normally stays in second position.

The sentence here has a basic, neutral order:

• Við | göngum | yfir þessa brú | saman

If you move another element to the front for emphasis, the verb still usually stays second:

• Yfir þessa brú göngum við saman.

That is normal Icelandic word order behavior.

No. Yfir can mean several things depending on context, including:

• over
• above
• across

In this sentence, across is the most natural English translation, because the idea is moving from one side of the bridge to the other.

So although yfir often corresponds to over, here across is better English.

You usually learn Icelandic nouns together with their gender. Brú is a feminine noun.

In this sentence, you can see that from the form of the demonstrative:

• þessa is the feminine singular accusative form of þessi

So the phrase þessa brú tells you that brú is feminine.

This is a good habit in Icelandic: learn nouns with gender and, when possible, with a full phrase rather than as isolated words.