Po lekcji córka pokazuje mi swój rysunek, a syn chowa kredki do piórnika.

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

• basic form: lekcja
• after po: po lekcji

So po lekcji means after the lesson/class.

A few similar examples:

• po pracy = after work
• po obiedzie = after lunch
• po filmie = after the film

Polish does not have articles like a or the.

So:

• córka can mean the daughter or a daughter
• syn can mean the son or a son

The exact meaning comes from context. If you want to make possession explicit, you can add a possessive:

• moja córka = my daughter
• mój syn = my son

But in this sentence, that is unnecessary because the context already makes it clear.

Both mi and mnie can mean to me, but they are used differently.

• mi is the normal, unstressed form in neutral sentences
• mnie is used for emphasis, contrast, or after prepositions

So in a plain sentence:

• córka pokazuje mi swój rysunek = the natural choice

If you wanted contrast, you might say:

• Mnie pokazuje swój rysunek, nie tobie = She is showing her drawing to me, not to you.

Because swój is the reflexive possessive. It is used when the owner is the same as the subject of the clause.

Here the subject is córka, so:

• córka pokazuje swój rysunek = the daughter shows her own drawing

Using jej rysunek would usually suggest some other girl's/woman's drawing, not the daughter's own drawing.

This is a very important Polish pattern:

• Anna czyta swoją książkę = Anna is reading her own book
• Anna czyta jej książkę = Anna is reading her book, but the book belongs to another female person

It is in the accusative, because it is the direct object of pokazuje.

The tricky part is that for many masculine inanimate nouns, the accusative singular looks exactly the same as the nominative singular.

So:

• nominative: rysunek
• accusative: rysunek

That is why the form does not visibly change.

Compare that with a masculine animate noun:

• nominative: syn
• accusative: syna

Kredki is the direct object of chowa, so it is in the accusative plural.

The noun is:

• singular: kredka
• plural: kredki

For feminine nouns, the nominative plural and accusative plural are often identical, so the form stays kredki.

So even though it looks like the basic plural form, its role in the sentence is accusative:

• chowa kredki = puts away the crayons / hides the crayons

Because the preposition do requires the genitive case.

The noun is:

• basic form: piórnik
• genitive singular: piórnika

So:

• do piórnika = into the pencil case / to the pencil case

This is a common pattern:

• do domu = into/to the house
• do szkoły = to school
• do torby = into the bag

This is the difference between movement and location.

• do piórnika = into the pencil case, showing movement toward the inside
• w piórniku = in the pencil case, showing location

So:

• syn chowa kredki do piórnika = the son is putting the crayons into the pencil case
• kredki są w piórniku = the crayons are in the pencil case

Both can mean and, but they are not always used in the same way.

• i is a straightforward and
• a often links two parallel or slightly contrasted pieces of information

Here the sentence compares what two different people are doing:

• the daughter does one thing
• the son does another

So a sounds very natural:

• córka pokazuje..., a syn chowa...

It is not a strong but, but it does create a mild contrast or side-by-side comparison.

Because the sentence describes the actions as ongoing, ordinary, or simply presented without focusing on completion.

• pokazywać = imperfective
• chować = imperfective

In Polish, perfective verbs usually do not have a true present-time meaning. Their present-looking forms normally refer to the future.

So:

• pokazuje = is showing / shows
• chowa = is putting away / puts away

But:

• pokaże = will show
• schowa = will put away / will hide

That is why imperfective is the natural choice here.

It can mean either, depending on context.

Polish present tense often covers both:

• an action happening right now
• a habitual or repeated action

So this sentence could describe:

• what is happening now after a lesson
• or a typical scene that happens after lessons

English usually forces you to choose between is showing and shows, but Polish often leaves that to context.

No, Polish word order is quite flexible.

The sentence begins with Po lekcji to set the time frame first:

• Po lekcji córka pokazuje mi swój rysunek, a syn chowa kredki do piórnika.

That is a very natural order, but other orders are possible if you want different emphasis.

For example:

• Córka po lekcji pokazuje mi swój rysunek...
• Swój rysunek córka pokazuje mi...

However, some words prefer certain positions. For example, mi is a short unstressed form, so it usually stays close to the verb and normally does not stand at the beginning of the sentence.

It is singular.

The basic noun is lekcja = lesson / class.

Its locative singular form is lekcji, so:

• po lekcji = after the lesson / after class

The form lekcji can also appear in other cases and sometimes in the plural, so the ending alone does not always tell you everything. In this sentence, the preposition po and the context show that it is locative singular.