Czwartego maja moja kuzynka zdaje egzamin na prawo jazdy.

Because Polish dates use the genitive case for both the day number and the month.

So:

• czwarty maj would just be something like fourth May in a dictionary-style or incorrect sentence context
• czwartego maja means on the fourth of May

This is the normal pattern for dates in Polish:

• pierwszego stycznia = on the first of January
• trzeciego lutego = on the third of February
• dwudziestego maja = on the twentieth of May

In full sentences, Polish often omits the word day and just uses this date form by itself.

Because month names also usually appear in the genitive when giving a date.

The base form is:

• maj = May

But in a date:

• czwartego maja = the fourth of May

Other examples:

• w maju = in May
• piątego maja = on the fifth of May

So maja here is not a different word; it is just the correct case form of maj.

Yes, in terms of meaning. English says on the fourth of May, but Polish often just says Czwartego maja without any separate word for on.

So:

• Czwartego maja moja kuzynka zdaje egzamin...
  means
• On the fourth of May, my cousin is taking an exam...

Polish often expresses time without a preposition when using dates in this structure.

Because moja kuzynka is the subject of the sentence, so it is in the nominative case, which is the basic dictionary form.

• moja = my
• kuzynka = female cousin

She is the person doing the action of zdaje.

If the cousin were the object instead, the form would change. For example:

• Widzę moją kuzynkę = I see my cousin

But here:

• moja kuzynka zdaje... = my cousin is taking...

Yes.

• kuzynka = female cousin
• kuzyn = male cousin

Polish often marks gender in nouns more clearly than English does. Since kuzynka is feminine, that is also why the possessive adjective is moja and not mój.

Zdaje is the 3rd person singular form of zdawać.

Here it means:

• to take an exam
• and depending on context, sometimes also to pass an exam

In this sentence, the most natural interpretation is:

• my cousin is taking the driving test

Why? Because the sentence gives a specific future date, so it sounds like an upcoming exam event.

A useful distinction:

• zdawać egzamin = to be taking/sitting an exam
• zdać egzamin = to pass an exam

So aspect matters:

• zdawać = imperfective
• zdać = perfective

Because zdaje egzamin often refers to the event of sitting/taking the exam, not necessarily the successful result.

Compare:

• Moja kuzynka zdaje egzamin jutro. = My cousin is taking the exam tomorrow.
• Moja kuzynka zda egzamin. = My cousin will pass the exam.

The first focuses on the exam as an event.
The second focuses on success.

So in your sentence, zdaje egzamin is the normal way to say someone is taking an exam on that date.

This phrase means driving test or more literally exam for a driving licence.

Breakdown:

• egzamin = exam
• na = for
• prawo jazdy = driving licence

So:

• egzamin na prawo jazdy = exam for a driving licence

This is just the standard Polish expression. Even though English might say driving test, Polish commonly phrases it with egzamin na prawo jazdy.

Literally, it means something like the right to drive.

• prawo = law, right
• jazdy = of driving / of travel by vehicle

But as a fixed phrase, prawo jazdy simply means driving licence or driver’s license.

So you should learn it as a whole expression.

In this expression, na is the normal preposition used with egzamin when the exam qualifies you for something or is aimed at getting something.

So:

• egzamin na prawo jazdy = driving test
• egzamin na studia can mean an entrance exam for university in some contexts

By contrast, egzamin z... often means an exam in a subject:

• egzamin z matematyki = exam in mathematics
• egzamin z polskiego = exam in Polish

So:

• na prawo jazdy = for a driving licence
• z matematyki = in mathematics

Because this is a fixed expression, and in modern Polish na prawo jazdy is the standard form.

Learners often expect na to force the accusative, and often it does. But fixed phrases and institutional expressions do not always feel fully transparent word by word anymore.

In practice, the safest thing is to learn:

• egzamin na prawo jazdy
• mieć prawo jazdy = to have a driving licence

Treat prawo jazdy as a set phrase.

Formally, zdaje is present tense, but in this sentence it refers to a future scheduled event.

Polish often uses the present tense for planned future events, just like English can:

• Tomorrow she takes the exam
• On May 4th my cousin takes the driving test

So:

• Czwartego maja moja kuzynka zdaje egzamin...
  means something like
• On the fourth of May, my cousin is taking the driving test

It is present in form, future in meaning.

Yes. Polish word order is more flexible than English because case endings show grammatical roles.

Your sentence:

• Czwartego maja moja kuzynka zdaje egzamin na prawo jazdy.

Other possible orders:

• Moja kuzynka czwartego maja zdaje egzamin na prawo jazdy.
• Egzamin na prawo jazdy moja kuzynka zdaje czwartego maja.

These versions may sound slightly different in emphasis, but the basic meaning stays the same.

The original sentence is very natural because it starts with the date, which sets the time frame immediately.

Because in Polish, month names are not capitalized in normal writing.

So:

• maj
• styczeń
• luty

Also, the date word here is just an ordinary ordinal form:

• czwartego = fourth

So the lowercase spelling in Czwartego maja is normal, except that Czwartego is capitalized here simply because it begins the sentence.

A natural answer would be:

• Czwartego maja. = On the fourth of May.

You could also hear:

• 4 maja in writing
• Czwartego maja zdaje egzamin na prawo jazdy.

In speech, the fully inflected form czwartego maja is the important thing to recognize and produce.