Þurrkunin hefst að því gefnu að rakamælingin sýni raka.

Yes. Að því gefnu að is a set phrase meaning provided that, given that, or on the condition that.

It is fairly formal and is common in written or technical language. In everyday speech, Icelandic often uses simpler wording such as ef for if.

So in this sentence, it introduces the condition under which the drying begins.

Because the two að words are doing different jobs.

• The first að is part of the fixed expression að því gefnu
• The second að is the conjunction that, introducing the subordinate clause

So the structure is roughly:

• að því gefnu = given that / on that condition
• að rakamælingin sýni raka = that the moisture measurement shows moisture

This double að is completely normal in Icelandic in this expression.

Hefst comes from the verb hefjast, which means to begin or to start.

The -st ending is the usual marker of the Icelandic middle voice. In many cases, this gives a meaning like:

• something begins by itself
• something happens
• something is set in motion

Compare:

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

So Þurrkunin hefst means the drying begins.

That -in is the definite article attached to the noun. Icelandic usually adds the onto the end of the word instead of putting a separate word in front.

So:

• þurrkun = drying
• þurrkunin = the drying

and

• rakamæling = moisture measurement
• rakamælingin = the moisture measurement

Both of these are feminine singular nouns in the nominative here.

Yes. Rakamælingin is a compound noun:

• raka- = moisture
• mæling = measurement

So rakamæling literally means moisture-measurement.

The raka- part is related to the noun raki meaning moisture. In compounds, Icelandic very often joins words together like this, and the first element may appear in a form that matches the genitive or a compound stem.

So yes: the raka- in rakamælingin and the final raka are from the same noun.

Because sýni is the present subjunctive form of sýna (to show), while sýnir is the present indicative.

In this sentence, the clause after að því gefnu að expresses a condition or assumption, and Icelandic commonly uses the subjunctive in that kind of formal conditional clause.

So:

• sýnir = shows
• sýni = show, in a subjunctive/conditional sense

That is why rakamælingin sýni raka is natural here.

Because raka is the accusative singular form of raki.

The noun raki means moisture. In this sentence it is the direct object of sýna (to show), so it appears in the accusative:

• nominative: raki
• accusative: raka

So:

• rakamælingin sýni raka = the moisture measurement shows moisture

Here is the case structure of the sentence:

• Þurrkunin — nominative
  Subject of hefst

• því — dative
  Used because of the first að in the fixed phrase að því gefnu

• gefnu — dative singular neuter
  Agrees with því

• rakamælingin — nominative
  Subject of sýni

• raka — accusative
  Direct object of sýni

This is a good example of how Icelandic case marking shows grammatical roles very clearly.

The given order is natural and clear, but it is not the only possible one.

You can also put the condition first:

Að því gefnu að rakamælingin sýni raka, hefst þurrkunin.

That still means the same thing. The difference is mainly one of style and emphasis.

The original version starts with the main action first:

• Þurrkunin hefst ... = the main point is introduced immediately

The reordered version starts with the condition first:

• Að því gefnu að ... = the condition is foregrounded

Both are correct.