Ef sjónprófið sýnir að ég þarf þau, læt ég gera ný gleraugu strax.

This is because of Icelandic V2 word order in main clauses.

The whole Ef-clause, Ef sjónprófið sýnir að ég þarf þau, takes the first position in the sentence. After that, the finite verb of the main clause must come second, so you get:

Ef ... , læt ég ...

not

Ef ... , ég læt ...

If you started with the main clause instead, then you would say:

Ég læt gera ný gleraugu strax ef sjónprófið sýnir að ég þarf þau.

No. Here láta + infinitive is a causative construction.

So láta gera means something like:

• have something done
• get something made

So læt ég gera ný gleraugu means that I arrange for new glasses to be made, usually by an optician or someone else, not that I personally make them.

Because after láta, Icelandic normally uses a bare infinitive.

So you say:

• láta gera
• not láta að gera

This is similar to how some English verbs take a bare infinitive, as in make someone do something rather than make someone to do something.

Here að means that and introduces a content clause:

sýnir að ég þarf þau
= shows that I need them

This is a different use of að from the infinitive marker meaning to. Icelandic uses the same word for both functions, so you have to tell from context.

In this sentence:

• að after sýnir = that
• no að before gera, because láta takes a bare infinitive

Þau refers to gleraugu.

Since gleraugu is grammatically neuter plural, the pronoun referring to it is also neuter plural:

• gleraugu = glasses
• þau = them

So ég þarf þau means I need them, where them means the glasses.

For the meaning glasses / spectacles, gleraugu is treated as a plural noun in Icelandic, much like English glasses.

So Icelandic says:

• ný gleraugu
• þau
• plural agreement

The singular glerauga is not normally used to mean one pair of glasses. So when talking about spectacles, learners should think of gleraugu as a plural item.

Because the adjective has to agree with gleraugu.

Here gleraugu is:

• neuter
• plural
• indefinite

So the correct strong adjective form is ný:

• ný gleraugu = new glasses

Compare:

• nýtt = singular neuter
• ný = plural neuter here
• nýju gleraugun = the new glasses, with weak adjective + definite noun

So ný is exactly the form you expect with an indefinite neuter plural noun.

Yes. In Icelandic, nominative and accusative are often identical in the neuter plural.

That means these forms can look the same whether they are subjects or objects.

In this sentence:

• þau is the object of þarf
• ný gleraugu is the object of gera

But the forms do not visibly change, because neuter plural nominative and accusative are often the same.

Because sjónprófið means the eye test.

The ending -ið is the suffixed definite article. Icelandic usually expresses the by attaching it to the noun:

• sjónpróf = an eye test / eye test
• sjónprófið = the eye test

In this sentence, it sounds like a specific test is meant, so the definite form is natural.

Icelandic often uses the present tense for future meaning when the context makes the timing clear.

So:

• Ef sjónprófið sýnir ...
• læt ég gera ...

can naturally mean:

• If the eye test shows ...
• I’ll have ... made

This is very common, especially in conditionals and planned future actions.

Þurfa normally takes a direct object, and that object is usually in the accusative.

Here, þau is the correct object form for gleraugu. One slightly tricky point is that in the neuter plural, nominative and accusative look the same, so þau does not visibly change.

So the important idea is:

• þurfa takes an object
• that object is þau
• the form happens to be the same as nominative in the neuter plural

Because Icelandic normally puts a comma between an initial subordinate clause and the main clause.

So this structure is standard:

Ef ... , læt ég ...

English punctuation can be a bit less strict in some cases, but in Icelandic this comma is very normal.

Yes, adverbs like strax can move, but the placement changes the emphasis a little.

In the given sentence, final position is neutral and natural:

... læt ég gera ný gleraugu strax

If you move it earlier, you usually highlight the immediacy more strongly:

• Strax læt ég gera ný gleraugu
• læt ég strax gera ný gleraugu

So the version in the sentence is a very normal, unmarked way to say it.