Po biegu mam gorące czoło i czerwony policzek od wiatru.

Because po in the sense of after takes the locative case in Polish.

• basic noun: bieg = run
• locative singular: biegu
• so: po biegu = after the run / after running

This is a very common pattern:

• po obiedzie = after lunch
• po pracy = after work
• po treningu = after training

So po bieg would be incorrect here.

Polish often uses mam + adjective + body part where English might prefer my ... is ....

So:

• mam gorące czoło literally = I have a hot forehead
• natural English meaning = my forehead is hot

This is very normal in Polish with body parts and physical states:

• Mam zimne ręce. = My hands are cold.
• Mam czerwone oczy. = My eyes are red.
• Mam suchą skórę. = My skin is dry.

You could build sentences with być in some contexts, but here mam sounds very natural.

Because Polish usually drops subject pronouns when the verb already shows the person clearly.

• mam already means I have
• so ja mam is only used when you want emphasis or contrast

For example:

• Mam gorące czoło. = neutral
• Ja mam gorące czoło, a on nie. = I have a hot forehead, but he doesn’t

So leaving out ja is the normal choice.

Because with body parts, Polish often leaves out the possessive if it is obvious whose body part is meant.

So mam gorące czoło naturally means my forehead is hot, even though my is not stated.

This is very common:

• Boli mnie głowa. = My head hurts.
• Umyłem ręce. = I washed my hands.
• Mam zamknięte oczy. = My eyes are closed.

You can add moje / mój for emphasis, but it is usually unnecessary.

They are in the accusative, because mam takes a direct object.

So the structure is:

• mam
  • co? = what do I have?

Answer:

• gorące czoło
• czerwony policzek

In this sentence, the accusative forms happen to look the same as the nominative forms:

• czoło is neuter singular, so nominative and accusative are the same
• policzek is masculine inanimate, so nominative and accusative are also the same

That is why you do not see any obvious noun ending change here.

Because adjectives in Polish must agree with the noun in gender, number, and case.

Here:

• czoło is neuter singular
  → adjective: gorące
• policzek is masculine singular
  → adjective: czerwony

So the endings are different because the nouns belong to different grammatical genders.

A few similar examples:

• zimne mleko = cold milk, neuter
• zimny wiatr = cold wind, masculine
• zimna kawa = cold coffee, feminine

Yes, and that is a good thing to notice.

Policzek is singular, so the sentence literally says one cheek. That is not grammatically wrong. It may suggest:

• only one cheek is red
• one cheek is being mentioned as an example
• the wind hit one side more strongly

If you want to say cheeks in general or both cheeks, you would say:

• czerwone policzki

So the sentence could be changed depending on the exact situation, but the singular form itself is fine.

Wiatru is the genitive singular form of wiatr.

That is because the preposition od normally takes the genitive.

So:

• wiatr = wind
• od wiatru = from the wind / because of the wind

This phrase expresses the cause of the redness.

Other examples:

• ból od stresu = pain from stress
• zaczerwienienie od słońca = redness from the sun
• łzy od dymu = tears from smoke

Not quite.

• od wiatru = because of the wind, as an effect caused by the wind
• na wietrze = in the wind / out in windy conditions

So in this sentence, od wiatru is about the cause of the red cheek.

Compare:

• Mam czerwony policzek od wiatru. = My cheek is red because of the wind.
• Stałem długo na wietrze. = I stood in the wind for a long time.

Przez wiatr is usually less natural for this kind of bodily effect.

Most naturally, it describes only czerwony policzek.

So the sentence is understood like this:

• Po biegu → explains the situation overall
• mam gorące czoło → hot forehead, likely from the run
• i czerwony policzek od wiatru → red cheek because of the wind

In other words, the final phrase attaches most naturally to the nearest idea, policzek.

If you wanted the wind to clearly describe both body parts, you would probably rephrase the sentence.

No, Polish word order is fairly flexible, but this version sounds natural and clear.

Po biegu mam gorące czoło i czerwony policzek od wiatru is a normal order because it starts with the time/context phrase po biegu and then gives the result.

You could rearrange parts for emphasis, but not every order sounds equally natural. For example:

• Mam po biegu gorące czoło... = possible, but less neutral
• Czerwony policzek mam od wiatru. = possible, but more contrastive or marked

So the original order is a good neutral model to learn from.

Because in Polish, adjectives normally come before the noun they describe.

So:

• gorące czoło
• czerwony policzek

That is the standard order.

Putting the adjective after the noun is possible in some special styles or expressions, but for ordinary everyday Polish, adjective + noun is the normal pattern.