Pojistník platí výši pojistného.

Not necessarily.

• pojistník = the policyholder, the person who has taken out the insurance contract
• pojištěný = the insured person, the person who is covered by the insurance

Sometimes these can be the same person, but they do not have to be. In insurance language, this distinction matters a lot.

In this sentence, pojistník is the person who pays.

Platí is the 3rd person singular present tense of platit.

So:

• já platím = I pay
• ty platíš = you pay
• on/ona/ono platí = he/she/it pays

Because pojistník is a singular subject, Czech uses platí.

Pojistník is the subject of the sentence, so it is in the nominative case.

The basic dictionary form is also pojistník, so here the subject appears in its base form.

In other words:

• Kdo platí? = Who pays?
• Pojistník = The policyholder

That is why it stays pojistník.

Because výši is in the accusative case.

The noun is:

• nominative: výše = amount, level, height

But in this sentence it is the direct object of platí:

• Pojistník platí co? = What does the policyholder pay?
• výši pojistného

For this feminine noun, the accusative singular form is výši.

So:

• výše = nominative
• výši = accusative

Because it depends on výše in the sense of amount of something.

Czech often uses the genitive after nouns like:

• výše = amount/level
• cena = price
• část = part
• druh = kind

So:

• výše pojistného = the amount of the premium

Here:

• výši = accusative, because it is the thing being paid
• pojistného = genitive, because it means of the premium

The basic form is pojistné.

In insurance language, pojistné means insurance premium. It looks like an adjective form, but here it functions as a noun. This is very common in Czech official or technical language.

Its forms include:

• nominative singular: pojistné
• genitive singular: pojistného

So in výši pojistného, the form pojistného means of the premium.

You can, and that is a very natural sentence.

There is a nuance:

• Pojistník platí pojistné. = The policyholder pays the premium.
• Pojistník platí výši pojistného. = The policyholder pays the amount of the premium.

The version with výši sounds more formal, administrative, or contract-like. It emphasizes the amount/level that is paid.

So the shorter sentence is more ordinary, while the longer one sounds more legal or bureaucratic.

No, Czech word order is relatively flexible, but changing it changes the emphasis.

The neutral order is:

• Pojistník platí výši pojistného.

Other possible orders:

• Výši pojistného platí pojistník.
  This emphasizes the amount of the premium
• Platí pojistník výši pojistného?
  This can sound like a question, depending on intonation

So the original sentence is the most neutral and standard order: subject + verb + object.

Czech does not have articles like English a and the.

So Czech simply says:

• Pojistník platí výši pojistného.

English needs to choose something like:

• The policyholder pays the amount of the premium
• or A policyholder pays the premium amount

In Czech, definiteness is understood from context, not from articles.

A simple approximate pronunciation is:

PO-yist-neek PLA-tee VEE-shee PO-yist-neh-ho

A few helpful points:

• poj sounds roughly like poy
• š sounds like sh
• í is a long ee
• č / š / ž are very important Czech sounds, though only š appears here

More carefully:

• pojistník ≈ poyist-neek
• platí ≈ pla-tee
• výši ≈ vee-shee
• pojistného ≈ poyist-neh-ho

The stress in Czech is usually on the first syllable of each word:

• PO-jistník
• PLA-tí
• VÝ-ši
• PO-jistného

No. Platit has more than one common meaning in Czech.

It can mean:

1. to pay

   • Platím účet. = I am paying the bill.

2. to be valid / to apply

   • To platí i dnes. = That still applies today.

In your sentence, the meaning is clearly to pay, because the object is výši pojistného.

It is understandable, but it sounds somewhat formal and administrative.

A more everyday sentence would often be:

• Pojistník platí pojistné.

The version with výši pojistného sounds like something from:

• an insurance contract
• terms and conditions
• a legal explanation
• an official document

So yes, it is correct, but it belongs more to formal written Czech than casual speech.