Egzamin z polskiego jest piątego maja, więc uczę się codziennie.

Egzamin z polskiego means an exam in Polish / a Polish-language exam.

In Polish, the pattern egzamin z + noun in the genitive is very common for school subjects:

• egzamin z matematyki = a math exam
• egzamin z historii = a history exam
• egzamin z polskiego = a Polish exam

Polski egzamin would usually sound like a Polish exam in the sense of an exam that is Polish, which is not the intended meaning here.

After the preposition z in this meaning, Polish uses the genitive case.

So:

• polski = basic dictionary form
• polskiego = genitive form

That is why you get:

• egzamin z polskiego
• lekcja z angielskiego
• sprawdzian z biologii

This is something you often just learn as a fixed pattern: z + genitive for school subjects after words like egzamin, test, sprawdzian.

In Polish, names of school subjects are often expressed with adjective forms used like nouns.

So:

• polski can mean Polish as an adjective
• polski / polskiego can also stand for the Polish language / the Polish subject, depending on context

That is why z polskiego naturally means in Polish language / in the Polish subject.

You will see the same pattern with:

• angielski → z angielskiego
• niemiecki → z niemieckiego

Piątego maja is in the genitive.

In Polish, when giving a date after jest in a sentence like this, the usual pattern is:

• jest + ordinal number in the genitive + month in the genitive

So:

• piąty maj is not correct for a date
• jest piątego maja = it is on the fifth of May / it’s May fifth

Here:

• piątego = fifth in the genitive
• maja = May in the genitive

In Polish, dates are often stated without a preposition after być:

• Egzamin jest piątego maja.
• literally: The exam is the fifth of May.

This is normal Polish.

You can also say:

• Egzamin jest 5 maja.

English usually needs on in this kind of sentence, but Polish often does not.

No, that would be incorrect.

If you use w, the form of the date has to work with that structure, and w piątego maja is not natural Polish.

The normal way here is simply:

• Egzamin jest piątego maja.

If you want a version with a preposition, Polish more naturally uses other structures, for example:

• Egzamin jest zaplanowany na piątego maja. = The exam is scheduled for May fifth.

But in your sentence, jest piątego maja is the standard simple form.

Uczyć się means to study / to learn.

This verb normally appears with się when it means to study/learn:

• uczę się = I study / I am studying
• uczysz się = you study
• uczymy się = we study

Without się, uczyć usually means to teach:

• Uczę polskiego. = I teach Polish.
• Uczę się polskiego. = I study Polish.

So się is essential here because it changes the meaning.

Polish does not form the present tense with a separate verb like to be plus -ing.

English:

• I study
• I am studying

Polish often uses one present-tense form for both:

• uczę się

So uczę się codziennie can mean:

• I study every day
• or, depending on context, I’m studying every day

You do not add a separate word like jestem here.

Więc means so, therefore, or thus.

In this sentence:

• Egzamin z polskiego jest piątego maja, więc uczę się codziennie.
• The Polish exam is on May fifth, so I study every day.

It shows a result or consequence:

• exam is coming
• therefore I study every day

A close alternative is dlatego:

• ..., dlatego uczę się codziennie.

Both are common, though więc is often the more direct equivalent of so.

Codziennie is an adverb meaning every day / daily.

It modifies the verb:

• uczę się codziennie = I study every day

Każdy dzień means every day as a noun phrase, but you would usually need a different structure:

• Uczę się każdego dnia. = I study every day.

So both are possible, but:

• codziennie is simpler and very natural
• każdego dnia is also correct, a bit more formal or literary in some contexts

Yes, Polish word order is fairly flexible because endings show grammatical relationships.

The neutral order here is:

• Egzamin z polskiego jest piątego maja, więc uczę się codziennie.

But other orders are possible for emphasis, for example:

• Piątego maja jest egzamin z polskiego, więc uczę się codziennie.
• Codziennie uczę się, bo egzamin z polskiego jest piątego maja.

The original sentence sounds natural and straightforward. A learner should probably use this kind of neutral order first.

In Polish, months are not capitalized, so:

• maj
• maja

Also, polski is not capitalized here because it refers to the language/subject, not a nationality in a special title.

So:

• z polskiego = correct
• piątego maja = correct

This is different from English, where May is capitalized.

A rough English-friendly pronunciation is:

eg-ZAH-min z pohl-SKYE-geh jest pyon-TEH-go MAH-ya, vyents oo-CHEH shyeh tso-JEN-nyeh

A few useful sound notes:

• z here sounds like z
• cz in uczę sounds roughly like ch in chop, but harder
• się is pronounced roughly shyeh
• ę in uczę and się is a nasal vowel, but in normal speech learners often approximate it and are still understood
• więc begins with a sound like vyen-ts

A more natural Polish rhythm would stress:

• egZAmin
• polSKIEgo
• piĄtego? Actually in Polish stress is usually on the second-to-last syllable: piąTEgo
• MAja
• uCZĘ się
• coDZIENnie

Usually, no, not in the same everyday sense.

Uczę się is the normal way to say I study / I’m learning in general, especially for school subjects.

Studiuję usually means:

• I am a university student
• or I study at university

So:

• Uczę się polskiego. = I’m learning Polish.
• Studiuję filologię. = I study philology at university.

In your sentence, uczę się codziennie is the natural choice.