Hún vill skrifa sínar eigin glósur.

After vilja (to want), Icelandic normally uses a bare infinitive:

• Hún vill skrifa = She wants to write

So English to write becomes simply skrifa here. You do not normally add að after vill in this kind of sentence.

Vill is the 3rd person singular present form of vilja (to want).

A few common forms are:

• ég vil = I want
• þú vilt = you want
• hún vill = she wants

So hún vill means she wants.

Because sínar is a reflexive possessive. It is used when the owner is the same person as the subject of the clause.

Here, the subject is hún (she), and the notes belong to that same person, so Icelandic uses sínar:

• Hún vill skrifa sínar glósur = She wants to write her own notes

If you said hennar instead, it would normally mean some other female person's notes, not the subject's own.

So the contrast is roughly:

• sínar = her own / the subject's
• hennar = her, but not reflexive

Yes, sínar already shows that the notes belong to the subject. Adding eigin makes the idea more emphatic:

• sínar glósur = her notes / her own notes
• sínar eigin glósur = her own notes, very clearly and emphatically

So eigin strengthens the meaning and highlights that they are her own notes, not someone else's.

It is the direct object of skrifa, so it is in the accusative.

However, there is a small wrinkle: in this phrase, the forms happen to look the same as the nominative in the plural. So you identify the case mainly from the sentence structure:

• hún = subject
• vill skrifa = wants to write
• sínar eigin glósur = the thing being written, so it is the object

That makes it accusative by function, even though the form does not visibly change here.

Because it has to agree with glósur in gender, number, and case.

Glósur is:

• feminine
• plural
• accusative here (because it is the object)

So the reflexive possessive must match that, giving sínar.

This kind of agreement is very important in Icelandic:

• the possessive/reflexive word matches the noun it goes with

It does agree with the noun, but in this particular form the correct agreeing form is simply eigin, so you do not see an obvious extra ending.

So yes, eigin is agreeing with glósur too; it just happens that the surface form here is still eigin.

That is fairly common in Icelandic: sometimes agreement is present grammatically even when the word does not look very different.

No. This reflexive possessive is mainly used with third-person subjects.

So:

• Hún vill skrifa sínar eigin glósur = She wants to write her own notes

But with I or you, you use the ordinary possessives instead:

• Ég vil skrifa mínar eigin glósur = I want to write my own notes
• Þú vilt skrifa þínar eigin glósur = You want to write your own notes

So sínar is correct here because the subject is hún.

This is the normal, neutral Icelandic order:

• Hún = subject
• vill = finite verb
• skrifa = infinitive
• sínar eigin glósur = object

So the pattern is basically:

Subject + finite verb + infinitive + object

That is the most straightforward way to say it.

Yes. Icelandic allows other word orders for emphasis, as long as the verb rules are respected.

For example, you could front the object for emphasis:

• Sínar eigin glósur vill hún skrifa.

That sounds more like:

• It is her own notes that she wants to write

But the original sentence is the most neutral and natural form for a learner to start with:

• Hún vill skrifa sínar eigin glósur.

Yes, very often it does. Because of sínar eigin, the sentence strongly focuses on the notes being her own.

Depending on context, that can imply things like:

• she wants to make her own notes rather than borrow someone else's
• she wants to write notes in her own way
• she wants to produce her own set of notes

So the grammar is doing more than just showing possession; it also adds a sense of contrast or emphasis.