Við stöflum flutningakössunum.

It is the present tense, first-person plural form of að stafla.

So við stöflum means we stack or we are stacking.

Because Icelandic changes the verb to match the subject.

The dictionary form is að stafla, but with við you need the we form, which is stöflum.

There is also a vowel change here:

• a often becomes ö before an ending with u
• the ending for we is -um

So:

• stafla → stöflum

This vowel change is very common in Icelandic and is often called u-umlaut.

Because Icelandic writes compounds as one word.

The base word is flutningakassi, meaning a moving box or packing box. It is made from:

• flutninga- = moving / transport
• kassi = box

Then Icelandic adds the grammatical ending to the end of the whole compound, not as a separate word.

So flutningakössunum is just a fully inflected form of the compound noun.

It is built into flutningakössunum.

In Icelandic, the definite article is usually attached to the noun instead of being a separate word like English the.

So:

• flutningakössum = dative plural, indefinite
• flutningakössunum = dative plural, definite

That means flutningakössunum already includes the idea of the moving boxes.

Because the verb að stafla takes a dative object.

So Icelandic says:

• stafla einhverju = stack something

This is one of those verb patterns you need to learn with the verb itself. English does not show this kind of case marking clearly, but Icelandic does.

That is why you get:

• Við stöflum flutningakössunum

with flutningakössunum in the dative plural.

This is another example of the same vowel change.

When kassi appears in a form with -um, the a changes to ö:

• kassi
• kössum
• kössunum

So inside the compound:

• flutningakassi
• flutningakössunum

This kind of stem change is very normal in Icelandic.

It can mean either one, depending on context.

The Icelandic present tense often covers both:

• a general present, like We stack the moving boxes
• an ongoing action, like We are stacking the moving boxes

If Icelandic wants to emphasize that the action is happening right now, it can use a construction with vera að:

• Við erum að stafla flutningakössunum

It is the first part of the compound noun.

It is related to flutningur, which has to do with moving, transport, or a move. In compounds, Icelandic often uses a linking form rather than the plain dictionary form.

So flutningakassi is literally something like moving-box or transport-box.

Usually, yes.

Even though stöflum already shows first-person plural, Icelandic normally still includes the subject pronoun in an ordinary statement:

• Við stöflum flutningakössunum

So Icelandic is not like languages where subject pronouns are regularly dropped in normal sentences.

Yes, but the given sentence is the most neutral order.

• Við stöflum flutningakössunum is the straightforward statement.

Icelandic main clauses normally follow a verb-second pattern, so if you move another element to the front, the verb still stays in second position:

• Núna stöflum við flutningakössunum = Now we are stacking the moving boxes

Other orders are possible, but they usually sound more marked, more literary, or more contrastive.