Ég hræri í súpunni á meðan hún sýður.

Because hræra often appears in the expression hræra í e-u, which means to stir in something or to stir something around.

So this sentence uses a very natural Icelandic pattern:

• hræra = stir, mix
• hræra í súpunni = stir in the soup / stir the soup around

This is a good example of how Icelandic verb patterns do not always match English word-for-word.

Because after í in this sentence, Icelandic uses the dative case.

The preposition í can take two different cases:

• accusative when there is movement into something
• dative when something is in something or happening within it

Here, the action is taking place in the soup, so Icelandic uses the dative:

• í súpunni = in the soup

By contrast, súpuna is the accusative definite form and would fit a different structure, such as a direct object.

It shows two things at once:

1. the noun is definite: the soup
2. the noun is in the dative singular

Icelandic usually puts the at the end of the noun instead of using a separate word.

For súpa:

• súpa = soup
• súpu = soup, dative singular
• súpunni = the soup, dative singular

So í súpunni literally means something like in soup-the, but in normal English that is just in the soup.

Because Icelandic nouns have grammatical gender, and súpa is a feminine noun.

That means pronouns referring back to it also use feminine forms:

• súpa = a feminine noun
• hún = she/it for a feminine noun

So in this sentence, hún means it, even though the form is the same word that can also mean she.

English mostly uses it for things, but Icelandic often uses hann, hún, or það depending on the noun’s gender.

Because sjóða is an irregular verb.

Its present-tense stem changes:

• sjóða = to boil
• sýður = boils / is boiling

So hún sýður means it boils or it is boiling.

This kind of vowel change is common in strong or irregular Icelandic verbs, so it is something learners usually just have to get used to and memorize.

Because Icelandic often uses the simple present where English uses the present progressive.

So:

• Ég hræri can mean I stir or I am stirring
• hún sýður can mean it boils or it is boiling

In context, the meaning is clearly ongoing, so Icelandic does not need a special am ... -ing form the way English does.

Á meðan means while or during the time that.

It introduces a time clause:

• á meðan hún sýður = while it is boiling

So the whole sentence describes one action happening during another action.

Yes, very often you can.

Both of these are natural:

• Ég hræri í súpunni á meðan hún sýður.
• Ég hræri í súpunni meðan hún sýður.

In many contexts, the difference is small. Learners can think of both as meaning while. You will hear both in real Icelandic.

Yes. You can move the while-clause to the front:

• Á meðan hún sýður, hræri ég í súpunni.

That still means the same thing.

This is useful because it shows an important Icelandic word-order rule: when something comes first, the finite verb of the main clause still comes early. That is why you get:

• hræri ég
• not ég hræri

after the fronted clause.

Possibly, but it is not exactly the same pattern.

• hræra í súpunni is a very common way to say stir in the soup / stir the soup around
• hræra súpuna uses súpuna as a direct object

Learners should especially notice that the sentence you were given uses the natural and very common expression hræra í e-u. Even if another structure may sometimes be possible, hræra í súpunni is the pattern to learn from this example.