Hún setti hárspennuna í töskuna og keypti svo nýja hárþurrku í búðinni á horninu.

They are both past tense, 3rd person singular forms.

• setja → setti = put
• kaupa → keypti = bought

Because the subject is hún = she, the verb is in the she form.

Because the subject stays the same.

Icelandic, like English, can avoid repeating the subject in a coordinated sentence:

• Hún setti ... og keypti ...
• She put ... and bought ...

So the second clause still means she bought, even though hún is not said again.

Here svo means then, after that, or next.

So it links the two actions in sequence:

1. she put the hair clip in the bag
2. then she bought a new hair dryer

It does not mean English so in the sense of therefore here.

This is a very important Icelandic pattern:

• í + accusative = movement into
• í + dative = location in

So:

• í töskuna = into the bag
  There is movement into the bag, so Icelandic uses the accusative.

• í búðinni = in the shop
  This describes where the buying happened, so Icelandic uses the dative.

Yes. Á works similarly:

• á + accusative = movement onto/to
• á + dative = location on/at

Here á horninu means on/at the corner, which is a location, so it uses the dative.

Those endings are part of the noun’s case form and often also include the attached definite article.

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

Examples from the sentence:

• hárspenna = a hair clip
  hárspennuna = the hair clip, accusative singular

• taska = a bag
  töskuna = the bag, accusative singular

• búð = a shop
  búðinni = the shop, dative singular

• horn = a corner
  horninu = the corner, dative singular

So these endings are not random: they show case, number, gender, and definiteness.

Because the adjective has to agree with the noun.

Here:

• hárþurrka is feminine
• it is singular
• it is the direct object of keypti, so it is in the accusative

That means the adjective nýr = new must also be in the matching form:

• nýja hárþurrku

So both words are in the form required for feminine singular accusative.

Usually you learn Icelandic nouns together with their gender.

In this sentence:

• hárspenna = feminine
• taska = feminine
• búð = feminine
• hárþurrka = feminine
• horn = neuter

Gender matters because it affects:

• adjective forms
• article endings
• case endings

A lot of nouns ending in -a are feminine, which helps, but the safest method is still to learn each noun with its gender.

Yes.

• hárspenna = hár
  • spenna
• hárþurrka = hár
  • þurrka

So they are built like English compound words such as hair clip and hair dryer.

A useful rule: in Icelandic compounds, the last part usually determines the gender and inflection of the whole word.

Because the sentence treats them differently in context.

• hárspennuna = the hair clip
• töskuna = the bag
• búðinni = the shop
• horninu = the corner

These are presented as specific, identifiable things.

But nýja hárþurrku means a new hair dryer, which introduces a new item rather than referring to one already known.

So Icelandic is making the same kind of definite/indefinite distinction that English often makes.

Yes, it is natural.

After og, the subject is omitted because it is still hún, so the verb keypti comes right away. Then svo follows it.

So:

• Hún setti ... og keypti svo ...

is perfectly normal Icelandic word order.

You could also see other variants in Icelandic, but this one is straightforward and idiomatic.