Hann lagar bindið sitt fyrir framan spegilinn.

Lagar is the 3rd person singular present tense of the verb að laga.

So:

• að laga = to fix, adjust, straighten
• ég laga = I fix
• þú lagar = you fix
• hann lagar = he fixes

Because the subject is hann (he), Icelandic uses lagar.

Here að laga does not necessarily mean repair in the strong sense of fixing something broken. In this context it means something more like:

• adjust
• straighten
• arrange

So Hann lagar bindið sitt is naturally understood as He adjusts/straightens his tie.

Bindið is the noun bindi with the definite article attached.

• bindi = tie
• bindið = the tie

Icelandic usually adds the to the end of the noun instead of using a separate word like English the.

So:

• bindi = a tie / tie
• bindið = the tie

This is a very common Icelandic question.

Sitt is a reflexive possessive, meaning the thing belongs to the subject of the sentence.

In this sentence:

• Hann = he
• sitt = his own

So bindið sitt means his own tie.

If you said bindið hans, that would usually mean his tie belonging to some other male person, not necessarily the subject.

So the contrast is roughly:

• Hann lagar bindið sitt. = He adjusts his own tie.
• Hann lagar bindið hans. = He adjusts his tie, meaning someone else’s tie.

Because sinn/sín/sitt must agree with the noun it describes in gender, number, and case.

The noun here is bindi, which is:

• neuter
• singular

So the correct form is sitt.

Very roughly:

• sinn = masculine singular
• sína = feminine singular / some plural forms depending on case
• sitt = neuter singular

Since bindið is neuter singular, Icelandic uses sitt.

It is in the accusative, because it is the direct object of the verb lagar.

He is doing the action to the tie, so the tie is the object.

With bindi, the nominative and accusative singular definite form happen to look the same:

• nominative: bindið
• accusative: bindið

So you cannot see the case difference from the form alone here, but grammatically it is accusative.

And sitt matches that noun in case as well.

Fyrir framan means in front of.

It is a very common Icelandic expression of location.

So:

• fyrir framan spegilinn = in front of the mirror

You can think of fyrir framan as a fixed phrase here.

Examples:

• fyrir framan húsið = in front of the house
• fyrir framan skólann = in front of the school

Because the sentence says the mirror, not just a mirror.

• spegill = mirror
• spegilinn = the mirror

Again, Icelandic usually attaches the definite article to the end of the noun.

Also, after fyrir framan, the noun is in the accusative, and the accusative singular definite form of spegill is spegilinn.

This happens because Icelandic nouns often change slightly when they are declined.

The basic dictionary form is:

• spegill = mirror

But in the accusative singular stem, it becomes spegil-, and then the definite ending is added:

• spegil
  • -inn = spegilinn

This kind of stem change is normal in Icelandic declension.

In this kind of usage, fyrir framan takes the accusative.

That is why we get:

• fyrir framan spegilinn

So even though the phrase expresses location in English, Icelandic still uses the accusative after this expression.

No, Icelandic word order is somewhat flexible, although the version you have is very natural.

The sentence begins with the subject:

• Hann lagar bindið sitt fyrir framan spegilinn.

But Icelandic could also move the prepositional phrase for emphasis:

• Fyrir framan spegilinn lagar hann bindið sitt.

That still means essentially the same thing, though the emphasis is a little different.

The original version is probably the most neutral and straightforward.

Yes, in a normal full sentence like this, the subject pronoun is included.

Icelandic usually does not drop subject pronouns the way some languages do.

So:

• Hann lagar bindið sitt. = He adjusts his tie.

Without hann, it would not be a normal complete sentence in this context.

Yes, depending on context, English may translate it as:

• He fixes his tie
• He adjusts his tie
• He is straightening his tie

Icelandic present tense often covers both a simple present and what English might express with is ...-ing, depending on the situation.

So if someone is standing there doing it right now, English would often naturally say:

• He is adjusting his tie in front of the mirror.