Po egzaminie nie ma się czego wstydzić, jeśli zrobiło się kilka błędów.

Because po meaning after takes the locative case.

• egzamin = nominative
• egzaminie = locative

So po egzaminie means after the exam. This is a very common pattern in Polish:

• po lekcji = after class
• po obiedzie = after dinner
• po pracy = after work

This is a very common Polish pattern:

nie ma czego + infinitive = there is nothing to ... / one has nothing to ...

Here the infinitive is wstydzić się = to be ashamed.

So:

• nie ma się czego wstydzić
  = literally something like there is nothing for oneself to be ashamed of = natural English: there is nothing to be ashamed of / you have nothing to be ashamed of

It is an impersonal, general statement, not aimed at one specific person only.

Because wstydzić się requires the genitive case.

You say:

• wstydzić się błędu = to be ashamed of a mistake
• wstydzić się czegoś = to be ashamed of something

So the pronoun must also be genitive:

• co = nominative/accusative
• czego = genitive

That is why czego is correct here.

Here się belongs to wstydzić się, not to ma.

So this is not the verb mieć się as in Jak się masz?

Polish often lets się move away from the infinitive, especially in expressions like this. So:

• nie ma się czego wstydzić
• nie ma czego się wstydzić

Both are possible, but the first one is very common and idiomatic.

Because this is an impersonal construction.

Polish often uses this kind of structure to mean:

• if one made a few mistakes
• if you made a few mistakes
• if some mistakes were made

It avoids naming a specific subject such as I, you, or we. That makes the sentence sound more general and less direct.

Compare:

• jeśli zrobiłem kilka błędów = if I made a few mistakes
• jeśli zrobiłaś kilka błędów = if you made a few mistakes
• jeśli zrobiło się kilka błędów = if a few mistakes were made / if one made a few mistakes

Because in this impersonal past construction, Polish uses the neuter singular past form.

So even though the meaning may refer to people in general, the verb appears in a form like:

• robiło się
• mówiło się
• myślało się
• zrobiło się

This does not mean the subject is actually neuter. It is just the normal shape of this impersonal pattern.

Because kilka requires the noun after it to be in the genitive plural.

So:

• błąd = mistake
• błędy = mistakes
• kilka błędów = a few mistakes

The same happens with many quantity words:

• kilka książek = a few books
• kilka pytań = a few questions
• kilka dni = a few days

Both are used.

• popełnić błąd is a bit more formal or careful in style
• zrobić błąd / zrobić błędy is very common in everyday speech

So the sentence is understandable and natural enough. A slightly more formal version would be:

• Jeśli popełniło się kilka błędów, po egzaminie nie ma się czego wstydzić.

Yes. Polish word order is fairly flexible.

For example, you could also say:

• Jeśli zrobiło się kilka błędów, po egzaminie nie ma się czego wstydzić.

This means the same thing. The original version simply starts with po egzaminie to frame the situation first: after the exam.

Because Polish often leaves the subject unstated when it is obvious, unimportant, or deliberately general.

In this sentence, the speaker is not talking about one named person. The idea is more like:

• you shouldn’t be ashamed
• one shouldn’t be ashamed
• there is no reason to be ashamed

That general tone is created by the impersonal forms:

• nie ma się czego wstydzić
• jeśli zrobiło się kilka błędów

So the lack of a subject is completely normal here.