Dóttir mín vill hoppa á trampólíninu eftir hádegi.

Both are possible, but they are not equally neutral.

• dóttir mín is the more normal, everyday way to say my daughter
• mín dóttir is possible, but it often sounds more emphatic, contrastive, or stylistically marked

So in a plain statement like this, dóttir mín is the most natural choice.

Mín is the possessive pronoun my.

It agrees with dóttir in:

• gender: feminine
• number: singular
• case: nominative

Here, dóttir is the subject of the sentence, so it is in the nominative singular. Because of that, the possessive must match it, giving mín.

A few related forms are:

• minn = masculine nominative singular
• mín = feminine nominative singular
• mitt = neuter nominative singular

So the form mín is there because dóttir is a feminine singular subject.

Because after vill (wants / wants to), Icelandic normally uses a bare infinitive.

So:

• vill hoppa = wants to jump

You do not say vill að hoppa here.

This is similar to how English uses a bare infinitive after some verbs like can or must, but with want, English uses to. Icelandic simply does not need an extra word here.

This is about how á works with case.

With á:

• dative usually indicates location / being on something
• accusative usually indicates movement onto something

In this sentence, hoppa á trampólíninu means something like:

• jump on the trampoline
• as in be on it and bounce there

So Icelandic treats the trampoline as the location, which is why it uses the dative: á trampólíninu.

If you wanted to say jump onto the trampoline, with movement toward it, you would normally use the accusative:

• hoppa á trampólínið

So the difference is roughly:

• á trampólíninu = on the trampoline
• á trampólínið = onto the trampoline

It shows two things at once:

• definiteness: the
• case/number/gender: here, dative singular neuter

The base noun is:

• trampólín = trampoline

In this sentence, after á with a location meaning, it becomes dative singular:

• á trampólíni = on a trampoline

When you add the definite article the, Icelandic usually attaches it to the end of the noun:

• á trampólíninu = on the trampoline

So -inu is not just one simple ending; it is part of the combined Icelandic ending that marks the trampoline in this grammatical role.

Literally, it means after midday / after noon, but in natural English it often corresponds to:

• in the afternoon
• after noon

So in context, the sentence can mean that she wants to jump on the trampoline after noon or in the afternoon.

This is a very normal Icelandic time expression.

Here eftir takes the accusative when it means after in time.

So hádegi is functioning as an accusative form here.

The reason this is not obvious is that hádegi has the same form in nominative and accusative singular, so nothing visibly changes.

A useful comparison:

• eftir matinn = after the meal
  Here you can clearly see the accusative form matinn

So in eftir hádegi, the preposition is still governing the accusative, even though the noun form does not visibly show it.

Because this is a common general time expression.

Icelandic often leaves out the article in expressions like this when speaking generally about time:

• eftir hádegi = after noon / in the afternoon

It is similar to how English often says after lunch, after school, or at noon without needing the.

So hádegi here is being used in a broad, idiomatic time-expression sense, not as a specifically identified the midday.

Yes. Icelandic word order is somewhat flexible, but in a main clause the finite verb usually stays in the second position.

The original sentence is:

• Dóttir mín vill hoppa á trampólíninu eftir hádegi.

You could also front the time expression:

• Eftir hádegi vill dóttir mín hoppa á trampólíninu.

Notice what happens:

• eftir hádegi moves to the front
• vill still stays in second position
• dóttir mín moves after the verb

That verb-second pattern is very important in Icelandic main clauses.

In a sentence like this, vill most naturally means wants to.

So:

• Dóttir mín vill hoppa... = My daughter wants to jump...

Depending on context, forms of vilja can sometimes also relate to willingness or intention, but for learners, want / want to is the best basic meaning here.

Also note that vill is the present singular form of vilja.