Optikeren måler pupilafstanden under synsprøven.

Because optikeren is the definite form of optiker.

• en optiker = an optician
• optikeren = the optician

In Danish, the definite article is often added to the end of the noun instead of being a separate word. That is a very common pattern:

• en bil = a car
• bilen = the car

So Optikeren måler ... means The optician measures ...

Måler is the present tense of the verb at måle = to measure.

A useful mini-pattern:

• at måle = to measure
• måler = measures / is measuring
• målte = measured
• har målt = has measured

In this sentence, måler simply means measures.

Because Danish very often makes compound nouns by joining words together.

Here, pupilafstanden can be broken down like this:

• pupil = pupil
• afstand = distance
• pupilafstand = pupillary distance / distance between the pupils
• pupilafstanden = the pupillary distance

So English might use several words, but Danish often prefers one compound noun.

For the same reason as optikeren: it is in the definite form.

• en pupilafstand = a pupillary distance
• pupilafstanden = the pupillary distance

In this sentence, it refers to a specific measurement, so the definite form is natural.

Here under means during, not physically underneath something.

So:

• under synsprøven = during the eye test

Danish under can mean both:

• under bordet = under the table
• under mødet = during the meeting

You have to understand it from context.

Because Danish usually expresses definiteness by adding the article to the end of the noun.

• en synsprøve = an eye test
• synsprøven = the eye test

So under synsprøven means during the eye test.

You would use den synsprøve only in more specific structures, for example with an added description:

• den synsprøve, jeg tog i går = the eye test that I took yesterday

Yes.

It is made from:

• syn = sight / vision
• prøve = test

So synsprøve literally means something like vision test or eye test.

This kind of word-building is extremely common in Danish.

Yes. This is the standard subject–verb–object pattern:

• Optikeren = subject
• måler = verb
• pupilafstanden = object
• under synsprøven = time phrase

So the structure is very natural: The optician measures the pupillary distance during the eye test.

Danish main clauses usually have the finite verb in second position, and this sentence follows that rule perfectly.

Yes, that is also correct.

If you move under synsprøven to the front, Danish keeps the verb in second position:

• Under synsprøven måler optikeren pupilafstanden.

That is a very important Danish word-order rule.
When something other than the subject comes first, the verb still stays second.

So:

• Optikeren måler pupilafstanden under synsprøven.
• Under synsprøven måler optikeren pupilafstanden.

Both are grammatical, just with different emphasis.

The å in måler is pronounced like a broad vowel, somewhat like the vowel in British more or law, though not exactly the same.

A rough guide:

• måler sounds approximately like MOH-ler

Also note:

• the r at the end of -er in Danish is often not pronounced very strongly
• Danish pronunciation is usually softer and more reduced than English learners expect

So it is best to listen to native audio if possible, but a rough starting point is MOH-ler.

No. Like English, pupil can also refer to a student in some contexts. But in modern Danish, elev is often more common for student/pupil in school contexts.

In this sentence, because of pupilafstanden and the context of an optician and an eye test, pupil clearly means the pupil of the eye.