Tjónamatið hefst þrátt fyrir að viðgerðarkostnaðurinn sé hár.

It has two parts:

• Main clause: Tjónamatið hefst
• Subordinate concessive clause: þrátt fyrir að viðgerðarkostnaðurinn sé hár

A simple way to parse it is:

• tjónamatið = subject of hefst
• hefst = main verb
• viðgerðarkostnaðurinn = subject of sé
• sé = verb in the subordinate clause
• hár = adjective describing viðgerðarkostnaðurinn

So the sentence says that one thing starts, even though another condition is true.

The ending -ið is the suffixed definite article, so it means the.

• tjónamat = damage assessment
• tjónamatið = the damage assessment

This is very typical in Icelandic: instead of a separate word for the, the article is usually attached to the noun.

Also, tjónamat is a neuter noun, and -ið is the neuter singular definite ending here.

Hefst is the 3rd person singular present form of hefjast, meaning to begin / to start.

So:

• hefjast = to begin
• hefst = begins / starts

This is an -st verb, which often corresponds to something intransitive or middle-voice-like. Compare:

• hefja = to begin/start something
• hefjast = to begin/start

In this sentence, tjónamatið hefst means the damage assessment begins.

Þrátt fyrir að is a set expression meaning:

• despite the fact that
• even though
• although

It introduces a full clause.

Compare:

• þrátt fyrir veðrið = despite the weather
• þrátt fyrir að veðrið sé slæmt = despite the fact that the weather is bad

So in your sentence, þrátt fyrir að introduces the idea even though the repair cost is high.

No. Here að is a conjunction, not the infinitive marker.

It introduces a full finite clause:

• að viðgerðarkostnaðurinn sé hár

That clause has:

• a subject: viðgerðarkostnaðurinn
• a finite verb: sé

By contrast, the infinitive að would be used before a verb in the infinitive, as in:

• að fara = to go
• að vinna = to work

So this að is closer to English that than to English to.

Sé is the present subjunctive of vera (to be).

• er = is, indicative
• sé = be, subjunctive

After expressions like þrátt fyrir að, Icelandic often uses the subjunctive in the subordinate clause. So sé is the standard choice here.

That is why you get:

• þrátt fyrir að viðgerðarkostnaðurinn sé hár

rather than:

• þrátt fyrir að viðgerðarkostnaðurinn er hár

For a learner, the main thing to remember is: after þrátt fyrir að, seeing a subjunctive verb is normal.

It is a compound noun:

• viðgerð = repair
• kostnaður = cost
• -inn = the

So:

• viðgerðarkostnaður = repair cost
• viðgerðarkostnaðurinn = the repair cost

The -ar- in the middle is a linking element, very common in Icelandic compounds. It often comes from an older or genitive form of the first noun.

Icelandic uses compounds very freely, so long words like this are extremely common.

Because the adjective has to agree with viðgerðarkostnaðurinn.

Viðgerðarkostnaðurinn is:

• masculine
• singular
• nominative

So the adjective also appears in the masculine singular nominative form:

• hár = high

Compare:

• masculine: kostnaðurinn er hár
• feminine: upphæðin er há
• neuter: gjaldið er hátt

So hár is there because kostnaður is masculine.

Because this is a subordinate clause introduced by að.

In a main clause, Icelandic normally has verb-second word order. But in subordinate clauses, the subject often comes before the verb more straightforwardly.

So:

• að viðgerðarkostnaðurinn sé hár

has the order:

• conjunction
• subject
• verb
• complement

That is perfectly normal subordinate-clause word order in Icelandic.

Yes. Very often, þrátt fyrir is followed directly by a noun phrase, and then it takes the accusative.

For example:

• þrátt fyrir háan kostnað = despite high cost
• þrátt fyrir veðrið = despite the weather

So your sentence could be rephrased with a noun phrase instead of a full clause, for example:

• Tjónamatið hefst þrátt fyrir háan viðgerðarkostnað.

The version with að + clause is more explicit: it says despite the fact that the repair cost is high, not just despite the high repair cost.