Codzienna nauka polskiego pomaga mi mówić coraz lepiej.

Because codzienna is an adjective, and it describes the noun nauka.

• codzienna nauka = daily learning/study
• codziennie is an adverb, meaning daily / every day

So:

• codzienna nauka polskiego = daily study of Polish
• Uczę się polskiego codziennie = I learn Polish every day

In this sentence, the speaker is talking about a thing, daily study, so Polish uses the adjective codzienna.

Because it agrees with nauka, which is:

• feminine
• singular
• nominative here, because it is the subject of the sentence

Polish adjectives must match the noun they describe in gender, number, and case.

So:

• codzienny = basic dictionary form
• codzienna = feminine singular form, matching nauka

That is why you get codzienna nauka.

Because the noun nauka often takes the genitive case for what is being learned or studied.

So:

• nauka czego? = study/learning of what?
• answer: polskiego

This is a very common pattern in Polish:

• nauka języka = learning of a language
• nauka historii = study of history
• nauka matematyki = study of mathematics

So polskiego is in the genitive because it depends on nauka.

Yes. Very often polskiego is understood as a shortened form of języka polskiego.

So these are closely related:

• nauka polskiego
• nauka języka polskiego

Both mean learning/studying Polish, but the shorter version is very common and natural.

Why does this work? Because in Polish, the language name can be understood from the adjective form alone in context.

Because in Polish, just like in English, one verb can be followed by an infinitive to show the action involved.

Here:

• pomaga = helps
• mówić = to speak

So:

• pomaga mi mówić = helps me speak / helps me to speak

Only pomaga is conjugated.
Mówić stays in the infinitive because it expresses the action that is being helped.

This is similar to English:

• It helps me speak better

not

• It helps me I speak better

Mi is the dative form of ja meaning to me / me.

The verb pomagać normally works with the dative for the person receiving help:

• pomagać komu? = to help whom?
• mi = to me

So:

• pomaga mi = helps me

Other examples:

• Pomagam ci = I help you
• To pomaga nam = This helps us

So in this sentence, mi is there because Polish says literally helps to me speak better, even though in natural English we just say helps me speak better.

Sometimes yes, but mi is the normal choice here.

• mi is the unstressed, very common short form
• mnie is the fuller, stronger form

In a neutral sentence like this, mi sounds most natural:

• Codzienna nauka polskiego pomaga mi mówić coraz lepiej.

You might use mnie for emphasis or contrast, for example:

• Tobie to nie pomaga, ale mnie pomaga.

Also, mi usually does not like to stand at the very beginning of a sentence, while mnie can.

Because mówić is imperfective, and it refers to the general ability or ongoing activity of speaking.

The sentence is about improving your speaking ability over time:

• mówić = to speak, to be able to speak, to speak in general

Powiedzieć is perfective and usually means to say something once / to finish saying something. That does not fit well here.

So:

• pomaga mi mówić coraz lepiej = helps me speak better and better

This is about a growing skill, not one completed act of saying something.

Coraz lepiej means better and better or increasingly well.

Breakdown:

• coraz = more and more / increasingly
• lepiej = better

So together:

• mówić coraz lepiej = to speak better and better

This is a very common Polish pattern:

• coraz więcej = more and more
• coraz szybciej = faster and faster
• coraz trudniej = more and more difficultly / harder and harder

Because after a verb like mówić, Polish needs an adverb, not an adjective.

• dobrze = well
• lepiej = better
• lepszy = better, but as an adjective describing a noun

Here, lepiej describes how someone speaks:

• mówić dobrze = to speak well
• mówić lepiej = to speak better

But lepszy would need a noun:

• lepszy uczeń = a better student
• lepsza metoda = a better method

So coraz lepiej is correct because it describes the manner of speaking.

Yes, Polish word order is fairly flexible, but some versions sound more natural than others.

The original sentence is very natural:

• Codzienna nauka polskiego pomaga mi mówić coraz lepiej.

You could also hear:

• Codzienna nauka polskiego pomaga mi coraz lepiej mówić.

That is also understandable and acceptable.

However, not every rearrangement sounds equally good. Polish has freedom, but speakers still follow natural rhythm and emphasis. In this sentence:

• Codzienna nauka polskiego works well as the subject at the beginning
• mi usually stays in a short, unstressed position
• coraz lepiej naturally stays close to mówić

So yes, word order can change, but the original version is a very good standard one.