Близорукость и астигматизм требуют новых линз.

Because they form a compound subject joined by и (and).

• близорукость = singular noun
• астигматизм = singular noun
• together: близорукость и астигматизм = myopia and astigmatism

In Russian, when two singular nouns are joined by и, the verb is usually plural:

• Мама и папа пришли. = Mom and Dad arrived.
• Близорукость и астигматизм требуют... = Myopia and astigmatism require...

So требуют is plural because the subject is understood as two things together.

Because the verb требовать normally takes the genitive case for the thing required.

So:

• требовать чего? = to require what?
• линз is the genitive plural of линзы / линза
• новых matches линз in case and number, so it is also genitive plural

That is why you get:

• требуют новых линз

and not:

• требуют новые линзы

A useful pattern to remember is:

• требовать времени = to require time
• требовать помощи = to require help
• требовать новых линз = to require new lenses

They are in the nominative case, because they are the subject of the sentence.

You can identify the subject by asking what requires new lenses?

• Что требует новых линз?
• Близорукость и астигматизм.

So both nouns are nominative singular, even though together they make the verb plural.

Here are the basic forms:

• близорукость — feminine noun
• астигматизм — masculine noun
• линза — feminine noun

A few helpful notes:

• Nouns ending in -ость are very often feminine:
  скорость, молодость, слабость, близорукость
• астигматизм is a masculine noun ending in a consonant, which is a very common masculine pattern.
• линз is not the dictionary form; the dictionary form is линза.

Yes, that is normal. Линз is the genitive plural form of линза.

The noun changes like this:

• nominative singular: линза
• nominative plural: линзы
• genitive plural: линз

For many feminine nouns ending in -а / -я, the genitive plural can have a zero ending or a shortened form. So линза → линз is a standard pattern.

Compare:

• берёза → берёз
• слеза → слёз
• линза → линз

Yes. Russian word order is more flexible than English word order.

The sentence as given:

• Близорукость и астигматизм требуют новых линз.

is a neutral, straightforward order: subject + verb + object.

But other orders are possible for emphasis, for example:

• Новых линз требуют близорукость и астигматизм.

This puts more emphasis on новых линз.

Even though the word order can change, the case endings still show the grammatical roles:

• близорукость и астигматизм = subject
• новых линз = object governed by требовать

Yes, it is a normal and common word for myopia / nearsightedness.

It is built from:

• близорукий = near-sighted
• близорукость = near-sightedness / myopia

The ending -ость often turns an adjective into an abstract noun, similar to English endings like -ness:

• молодой → молодость = young → youth
• слабый → слабость = weak → weakness
• близорукий → близорукость = near-sighted → near-sightedness

So this is a very useful word-formation pattern to recognize.

The stress is:

близору́кость и астигмати́зм тре́буют но́вых линз

A rough pronunciation guide:

• близору́кость = blee-zah-ROO-kəstʹ
• астигмати́зм = ə-steeg-mə-TEEZM
• тре́буют = TRE-boo-yut
• но́вых = NO-vykh
• линз = leenz

A few pronunciation tips:

• Unstressed о often sounds closer to a or a reduced vowel.
• ых in новых has the Russian ы sound, which does not exist in English.
• линз ends in a consonant cluster, so say it cleanly: линз, not линзы.

By itself, линзы just means lenses. The exact meaning depends on context.

It can refer to:

• eyeglass lenses
• contact lenses
• sometimes lenses more generally

In a medical or vision context like this sentence, English may translate it as new lenses, and the precise type would be understood from the situation.

If Russian wants to be more specific, it can say:

• контактные линзы = contact lenses
• линзы для очков = lenses for glasses

Because Russian usually uses the singular when talking about a condition or disorder in general.

So:

• близорукость = myopia, as a condition
• астигматизм = astigmatism, as a condition

This is similar to English, where we also usually say:

• Myopia requires correction
• Astigmatism causes blurred vision

If you were talking about multiple cases among different people, you might use other wording, but in general statements about the conditions themselves, the singular is the natural choice.