Dóttir mín safnar skeljum í sandinum meðan ég horfi á bylgjurnar.

Both are possible, but they do not feel exactly the same.

• dóttir mín is the more neutral and very common way to say my daughter
• mín dóttir can sound more emphatic, contrastive, or a bit more literary/poetic, depending on context

So in an ordinary sentence like this, dóttir mín is the most natural choice.

Also note that mín agrees with dóttir:

• dóttir = feminine singular nominative
• mín = the matching possessive form

Because the verb að safna takes the dative case.

So:

• skeljar = nominative/accusative plural, which would not fit here
• skeljum = dative plural, which is what safna requires

This is something you simply have to learn with the verb:

• að safna einhverju = to collect something

So:

• hún safnar skeljum = she collects shells

Skeljum is the dative plural of skel (shell).

A learner often notices the ending -um, because that is a very common dative plural ending in Icelandic.

So here:

• singular: skel
• plural nominative/accusative: skeljar
• plural dative: skeljum

Since að safna needs the dative, skeljum is the correct form.

Both are grammatically possible, but they mean slightly different things in terms of definiteness.

• í sandi = in sand
• í sandinum = in the sand

The ending -num contains the definite article, so sandinum means the sand.

Here the sentence is talking about a specific place, so í sandinum is very natural.

Because í can take either the accusative or the dative, depending on meaning.

A very useful rule is:

• location / being in a place → dative
• motion into a place → accusative

Here the daughter is already in the sand, collecting shells there. That is location, so we use the dative:

• í sandinum = in the sand

Compare:

• hún er í sandinum = she is in the sand / in the sand area
• hún fer í sandinn = she goes into the sand

Because the verb is að horfa á, which means to look at or to watch.

In English, we also need a preposition in many cases:

• look at the waves

Icelandic works similarly here:

• horfa á eitthvað = look at something / watch something

So you cannot normally just say horfa bylgjurnar. The preposition á is part of the usual pattern of the verb.

Bylgjurnar means the waves.

Breakdown:

• bylgja = wave
• bylgjur = waves
• bylgjurnar = the waves

The ending -nar is the definite article attached to the plural noun.

So:

• á bylgjurnar = at the waves

It is accusative plural definite.

That is because horfa á normally takes á + accusative when it means look at / watch.

So:

• á bylgjurnar = at the waves
• bylgjurnar is the correct object form after á here

For this noun, the plural definite nominative and accusative happen to look the same, but in this sentence the function is accusative because it follows á with horfa á.

Meðan means while.

It connects two actions that happen at the same time:

• Dóttir mín safnar skeljum í sandinum
• meðan ég horfi á bylgjurnar

So the idea is:

• My daughter collects shells in the sand while I watch the waves.

It is a very common conjunction for simultaneous actions.

Because horfi is the correct 1st person singular present tense form of að horfa.

Present tense of að horfa:

• ég horfi = I watch / I am watching
• þú horfir
• hann / hún / það horfir

So:

• ég horfi á bylgjurnar = I watch the waves

The infinitive is horfa, but after ég you need the finite verb form horfi.

Safnar is the 3rd person singular present tense of að safna.

So:

• ég safna = I collect
• þú safnar
• hann / hún / það safnar

Since the subject is dóttir mín (my daughter), Icelandic uses the 3rd person singular:

• Dóttir mín safnar = My daughter collects / is collecting

It can be understood either way, depending on context.

The Icelandic present tense often covers both:

• My daughter collects shells in the sand while I watch the waves
• My daughter is collecting shells in the sand while I am watching the waves

Without more context, the sentence could describe:

1. a scene happening now, or
2. a habitual/repeated situation

That is very normal in Icelandic.

Yes. That would also be natural.

• ég horfi á bylgjurnar = simple present, often enough by itself
• ég er að horfa á bylgjurnar = more explicitly I am watching the waves right now

So:

• Dóttir mín safnar skeljum í sandinum meðan ég horfi á bylgjurnar.
• Dóttir mín safnar skeljum í sandinum meðan ég er að horfa á bylgjurnar.

Both work. The version with er að puts more emphasis on the ongoing action.

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

So:

• sandur = sand
• sandurinn / sandinum = the sand in different cases

And:

• bylgjur = waves
• bylgjurnar = the waves

That is why you do not see a separate word for the in this sentence. It is built into the noun forms themselves.