Umráðamaðurinn athugar taumskylduna rétt áður en hann undirritar ættleiðingarsamninginn.

Because Icelandic usually puts the definite article on the end of the noun instead of using a separate word like English the.

In this sentence:

• umráðamaðurinn = the umráðamaður
• taumskylduna = the taumskylda
• ættleiðingarsamninginn = the adoption agreement

This is one of the first big differences English speakers notice in Icelandic.

Umráðamaðurinn is the subject of the sentence, so it is in the nominative singular definite.

Its dictionary form is umráðamaður.
Then Icelandic adds the definite article:

• umráðamaður = an umráðamaður
• umráðamaðurinn = the umráðamaður

This noun is masculine, and nouns ending in -maður are very common in Icelandic.

Because taumskylduna is the direct object of athugar, and athuga takes the accusative.

The basic form is taumskylda.
Its accusative singular is taumskyldu.
With the definite article added, it becomes taumskylduna.

So the ending tells you two things at once:

• it is accusative
• it is definite (the ...)

Because this noun is also a direct object, this time of undirritar, so it is in the accusative singular definite.

The dictionary form is ættleiðingarsamningur.
Its forms look like this:

• ættleiðingarsamningur = nominative singular
• ættleiðingarsamning = accusative singular
• ættleiðingarsamninginn = accusative singular definite

So -urinn would be the nominative definite form, but here the sentence needs the accusative form -inn.

Áður en means before when it introduces a clause.
Adding rétt makes it stronger: right before or just before.

So:

• áður en = before
• rétt áður en = right before / just before

Here rétt is an adverb, not the adjective meaning correct.

Because áður en here introduces a full subordinate clause with its own subject and finite verb:

• hann = subject
• undirritar = finite verb

English often allows something like before signing, but Icelandic very often uses a full clause in situations like this:

• áður en hann undirritar ... = before he signs ...

That structure is very natural in Icelandic.

Because the second part is a separate clause, and Icelandic normally states the subject explicitly there.

So the structure is:

• main clause: Umráðamaðurinn athugar taumskylduna
• subordinate clause: áður en hann undirritar ættleiðingarsamninginn

The pronoun hann refers back to umráðamaðurinn.
Even though English also does this, learners sometimes expect Icelandic to omit the pronoun more often than it actually does.

Both athugar and undirritar are 3rd person singular present tense forms:

• athuga → athugar
• undirrita → undirritar

Icelandic often uses the present tense for:

• habitual actions
• general statements
• step-by-step descriptions
• vivid narrative style

So even if the English meaning feels like a sequence of actions, the present tense is perfectly normal here.

It is a compound noun, which Icelandic uses all the time.

You can break it down like this:

• ættleiðing = adoption
• -ar- = linking/genitive element
• samningur = agreement/contract
• -inn = the definite article in this case/form

So ættleiðingarsamninginn is literally something like the adoption-agreement.

Long Icelandic words are often much less scary once you split them into parts.

Here are the most useful dictionary forms:

• umráðamaðurinn → umráðamaður
• athugar → athuga
• taumskylduna → taumskylda
• undirritar → undirrita
• ættleiðingarsamninginn → ættleiðingarsamningur

This is useful because Icelandic sentences often contain inflected forms that look quite different from the dictionary entry.

This is a normal and natural word order, but Icelandic word order is somewhat flexible because of its case system.

The sentence as given is:

• Umráðamaðurinn athugar taumskylduna rétt áður en hann undirritar ættleiðingarsamninginn.

You could also front the time clause for emphasis. If you do that, Icelandic still follows the V2 rule in the main clause, so the finite verb comes early:

• Rétt áður en hann undirritar ættleiðingarsamninginn athugar umráðamaðurinn taumskylduna.

So yes, the order can change, but not randomly. Icelandic has patterns behind that flexibility.