Tryggingafélagið svarar þrátt fyrir að tjónsnúmerið vanti.

Both are typical Icelandic compounds:

• trygging = insurance
• félag = company, association
• -ið = the

So Tryggingafélagið means the insurance company.

And:

• tjón = damage, loss, claim
• númer = number
• -ið = the

So tjónsnúmerið means the claim number.

Icelandic very often builds meanings by combining words into one long compound.

Because Icelandic usually adds the definite article (the) to the end of the noun instead of using a separate word.

So:

• félag = company
• félagið = the company

and:

• númer = number
• númerið = the number

The same thing is happening in both Tryggingafélagið and tjónsnúmerið.

Only because it is the first word of the sentence.

Common nouns in Icelandic are not normally capitalized just because they are nouns. So in the middle of a sentence you would usually write tryggingafélagið, unless it were part of an official name.

svarar is the 3rd person singular present tense of svara (to answer / reply / respond).

So:

• ég svara = I answer
• þú svarar is not correct
• hann/hún/það svarar = he/she/it answers

Here the subject is Tryggingafélagið, which is singular, so svarar is the correct form.

Also, Icelandic present tense often covers both English answers and is answering/responding, depending on context.

þrátt fyrir að is a fixed expression meaning something like:

• although
• even though
• despite the fact that

It introduces a subordinate clause.

So in this sentence:

• Tryggingafélagið svarar = the insurance company replies
• þrátt fyrir að tjónsnúmerið vanti = even though the claim number is missing

A very literal feel is despite the fact that...

Because here þrátt fyrir is followed by a whole clause, not just a noun phrase.

Compare:

• þrátt fyrir rigninguna = despite the rain
• þrátt fyrir að það rigni = despite the fact that it is raining

So:

• þrátt fyrir
  • noun phrase
• þrátt fyrir að
  • clause

That is why að appears in your sentence.

vanti is the subjunctive form of vanta.

After expressions like þrátt fyrir að, Icelandic often uses the subjunctive, especially in more formal or written language. It gives the clause a more subordinate, less directly asserted feel.

So:

• vantar = indicative
• vanti = subjunctive

In everyday speech, you may also hear vantar in similar sentences, but vanti is very natural in a formal written sentence like this one.

vanta can mean several related things, depending on context:

• to be missing
• to be lacking
• to be needed

Here the natural meaning is to be missing / not be available.

So tjónsnúmerið vanti means the claim number is missing.

One useful detail: because tjónsnúmerið is a neuter singular definite form, its nominative and accusative look the same, so this form alone does not clearly show the case pattern. For a learner, the important thing here is simply the meaning: the claim number is missing.

Yes.

The main clause is straightforward:

• Tryggingafélagið svarar
  subject + verb

Then comes the subordinate clause:

• þrátt fyrir að tjónsnúmerið vanti

Inside that clause, tjónsnúmerið comes before vanti, which is normal.

A useful thing to know is that Icelandic is a V2 language in main clauses, so if you moved the subordinate clause to the front, the main clause would still keep the finite verb in second position:

• Þrátt fyrir að tjónsnúmerið vanti svarar tryggingafélagið.

That is also good Icelandic.

Yes. A common alternative would be:

• Tryggingafélagið svarar þótt tjónsnúmerið vanti.

or

• Tryggingafélagið svarar þó að tjónsnúmerið vanti.

These also mean although / even though the claim number is missing.

The difference is mostly one of style:

• þrátt fyrir að can feel a bit more explicit or formal
• þótt / þó að is often shorter and more everyday

þ is pronounced like the th in thin, thing, or thank.

So it is:

• þ = voiceless th sound

It is not like the th in this or that.

That means þrátt starts with the same kind of sound as English thin, not this.