Ég skrifa ekki undir samninginn fyrr en ég les hann vandlega.

Because skrifa undir is a fixed Icelandic expression. Literally, it means write under, but idiomatically it means sign a document.

So in this sentence:

• Ég skrifa ekki undir samninginn ... = I am not signing the contract ...

A lot of learners notice that this feels similar to English phrasal verbs. That is a useful way to remember it: skrifa undir works as one unit in meaning, even though its parts can be separated in the sentence.

This happens because ekki normally comes after the finite verb in a main clause, and undir is a particle belonging to the verb phrase skrifa undir.

So the structure is:

• Ég = I
• skrifa = write / sign
• ekki = not
• undir = particle completing the meaning sign

That is why you get:

• Ég skrifa ekki undir ...

rather than keeping skrifa undir together.

This is very natural in Icelandic with verb-particle combinations.

Because samninginn is the accusative singular definite form of samningur (contract, agreement).

Here it is the direct object of skrifa undir, so it appears in the accusative:

• nominative: samningur = a contract
• accusative: samning = a contract
• accusative definite: samninginn = the contract

The -inn ending is the attached definite article, so samninginn means the contract.

Hann refers back to samninginn.

So:

• samninginn = the contract
• hann = it

Even though hann literally looks like he, Icelandic grammatical gender is not the same as natural gender. The noun samningur is masculine, so the pronoun used for it is hann.

In English we say it, but in Icelandic nouns have grammatical gender, so the pronoun has to match that gender.

Because the pronoun is the object of les (read), so it must be in the accusative case.

For the masculine singular pronoun:

• nominative: hann = he
• accusative: hann = him / it
• dative: honum
• genitive: hans

In this sentence, hann means it, referring to the contract, and it is in the accusative because it is what is being read.

Fyrir en and fyrr en are different, and here the correct phrase is fyrr en.

In this sentence, fyrr en means before or until in the sense of not until.

So:

• Ég skrifa ekki undir samninginn fyrr en ... = I won’t sign the contract until / before ...

Because the main clause is negative (ekki), English often translates this most naturally as not until.

So the idea is:

• I am not signing the contract until I read it carefully.

Icelandic often uses the present tense to talk about future actions, especially when the future is clear from context.

So:

• Ég skrifa ekki undir ...
• ... fyrr en ég les hann ...

can naturally refer to a future situation, even though both verbs are grammatically present tense.

English can do something similar:

• I’m not signing it until I read it.

Also, after time words like fyrr en, Icelandic commonly uses the present tense where English may also use a present form.

Les is the 1st person singular present tense of lesa (to read).

So:

• að lesa = to read
• ég les = I read / I am reading

In the sentence:

• ... fyrr en ég les hann vandlega = ... until I read it carefully

It matches ég, the subject.

Vandlega means carefully, and it is an adverb.

It describes how the contract is being read:

• les hann vandlega = read it carefully

A useful pattern to notice is that many Icelandic adverbs end in -lega, somewhat like English adverbs often end in -ly.

That is a natural position for an adverb of manner in Icelandic. It modifies the verb phrase:

• ég les hann vandlega

This is similar to English:

• I read it carefully

The adverb often comes after the object, though word order can vary somewhat depending on emphasis and style.

Yes. Undirrita also means to sign.

So a more formal or written alternative could be:

• Ég undirrita ekki samninginn fyrr en ég les hann vandlega.

But skrifa undir is very common and natural in everyday language. A learner should know both, but the sentence you were given uses the more everyday expression.

Because Icelandic does not need a separate word equivalent to English will in every future situation.

Very often, simple present tense is enough when the meaning is clearly future from context.

So Icelandic can say:

• Ég skrifa ekki undir ...

where English may prefer:

• I won’t sign ...
• I’m not signing ...

Both are natural translations depending on context.

Most naturally, it sounds like a firm decision or condition:

• I will not sign the contract until I read it carefully.

The speaker is setting a condition before signing. The structure:

• ekki ... fyrr en ...

is very useful for expressing this kind of idea:

• not ... until ...
• won’t ... before ...

So beyond the individual words, this is a good sentence pattern to learn as a whole.