Ég ætla að láta lita hárið mitt á morgun.

This is a very common Icelandic pattern.

Láta + infinitive means to have something done, to get something done, or to cause something to happen.

So here:

• láta = have/arrange/cause
• lita = dye, color

Together, láta lita means have dyed rather than simply dye.

Not really.

Láta can mean different things in different contexts, including let, make, leave, or have something done. In this sentence, it has the causative meaning:

• Ég læt lita hárið mitt = I have my hair dyed
• not I let dye my hair

So in this context, the best English idea is have or get something done.

Because the two verbs follow different patterns.

• ætla að + infinitive is the normal construction
  
  • Ég ætla að fara
  • Ég ætla að læra
• láta + bare infinitive is also a normal construction
  
  • láta gera
  • láta laga
  • láta lita

So Ég ætla að láta lita... is exactly what you would expect:

• ætla að
• then láta
• then the bare infinitive lita

You do not say láta að lita here.

Because after láta, Icelandic uses the infinitive.

So:

• láta lita hárið mitt = have my hair dyed

Even though English uses dyed, Icelandic does not use the past participle here.

A useful comparison:

• láta lita hárið mitt = have my hair dyed
• fá hárið mitt litað = get my hair dyed

With fá, you often get litað. With láta, you use lita.

Both can exist, but hárið mitt is very natural here.

A few things are happening:

1. hárið is the definite form of hár

   • hár = hair
   • hárið = the hair

2. Icelandic very often puts the possessive after the noun

   • hárið mitt = my hair

3. When the possessive comes after the noun, the noun is usually in the definite form

   • so hárið mitt, not just hár mitt

Mitt hár is possible, but it can sound more marked, contrastive, or stylistically different depending on context. In everyday speech, hárið mitt is extremely normal.

Because possessive pronouns in Icelandic agree with the noun in gender, number, and case.

The noun hár is neuter singular, so the possessive must also be neuter singular:

• masculine: minn
• feminine: mín
• neuter: mitt

Since hár is neuter, you say mitt:

• hárið mitt

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

However, for this noun, the nominative and accusative forms look the same in the singular definite:

• nominative: hárið
• accusative: hárið

So the case is accusative because of its role in the sentence, even though the form does not visibly change.

Usually, no.

This sentence normally means that someone else will do it for you, for example at a salon.

• Ég ætla að láta lita hárið mitt á morgun
  = I’m going to have my hair dyed tomorrow

If you mean that you yourself will dye it, you would normally say:

• Ég ætla að lita hárið mitt á morgun

So the presence of láta is important.

It mainly expresses intention or plan.

So Ég ætla að... is often like:

• I intend to...
• I’m going to...

In many everyday situations, it works a lot like English be going to, but the core idea is that the speaker has a plan or intention.

So here, the sentence is not just stating a future fact. It suggests a plan:

• I’m planning to have my hair dyed tomorrow

Because á morgun is the normal Icelandic expression for tomorrow.

It is best learned as a fixed phrase:

• á morgun = tomorrow

Even though English does not use a preposition here, Icelandic does. That kind of mismatch is very common in time expressions across languages.

Yes.

The version you have is very natural:

• Ég ætla að láta lita hárið mitt á morgun

But you can also front the time expression:

• Á morgun ætla ég að láta lita hárið mitt

That puts more focus on tomorrow.

Notice the word order in the second version:

• when Á morgun comes first,
• the finite verb ætla comes next,
• and then the subject ég

That is a normal example of Icelandic V2 word order.

Yes.

Lita basically means to color, to dye, or to tint, depending on context. So it can be used for hair, fabric, objects, and more.

In this sentence, because the object is hair, the most natural English meaning is dye.