Po škole si děti hrají na hřišti za domem.

Because po requires the locative case here.

• škola = basic dictionary form, school
• škole = locative singular

In time expressions, po + locative means after:

• po škole = after school
• po obědě = after lunch
• po práci = after work

So po škole is grammatically required.

Here si is part of the verb expression hrát si, which means to play in the sense of children playing / having fun.

Compare:

• děti si hrají = the children are playing
• hrají fotbal = they are playing football
• hraje na kytaru = he/she plays the guitar

So in this sentence, si is not really translated as a separate English word. It is just part of how Czech says to play in this meaning.

Because si is a clitic in Czech. Clitics usually go in the second position in the sentence or clause.

In:

Po škole si děti hrají...

the first unit is Po škole, so si comes right after it.

This is very normal Czech word order.
A sentence like Po škole děti si hrají sounds unnatural in standard Czech.

You can also move other parts around, but si still tends to stay in second position:

• Děti si po škole hrají na hřišti.

Děti is nominative plural, because it is the subject of the sentence.

• singular: dítě = child
• plural: děti = children

This is an irregular plural, so it does not look like a simple ending change.

In this sentence:

• děti = the children
• they are the ones doing the action

Because the subject děti is plural, so the verb must also be 3rd person plural.

The verb is hrát = to play.

Present tense:

• hraju / hraji = I play
• hraješ = you play
• hraje = he/she/it plays
• hrajeme = we play
• hrajete = you (plural) play
• hrají = they play

So:

• děti hrají = the children play

Because na with a location takes the locative case.

• hřiště = playground
• na hřišti = on/at the playground

Here the children are already there, so Czech uses na + locative.

Compare:

• Jsou na hřišti. = They are at the playground.
• Jdou na hřiště. = They are going to the playground.

So:

• na hřišti = location
• na hřiště = movement toward it

Domem is instrumental singular of dům (house).

The preposition za changes case depending on meaning:

• za + instrumental = location, behind
• za + accusative = movement, to behind / for / beyond depending on context

Here it is a location:

• za domem = behind the house

Compare:

• Děti si hrají za domem. = The children are playing behind the house.
• Běží za dům. = They are running behind the house.

So the sentence uses instrumental because it describes where they are.

Yes. Czech word order is quite flexible, because cases show grammatical relationships.

This sentence is a natural neutral version:

• Po škole si děti hrají na hřišti za domem.

But other orders are possible, for example:

• Děti si po škole hrají na hřišti za domem.
• Na hřišti za domem si děti po škole hrají.

The meaning stays similar, but the focus or emphasis changes.
The original version sounds very natural if you want to start with the time expression after school.

Because Czech does not have articles like a or the.

So Czech simply says:

• děti = children / the children
• hřiště = a playground / the playground
• dům = a house / the house

Whether English needs a or the depends on the context and translation, but Czech usually leaves that unstated.

Usually po škole means after school in the general time sense: after the school day or after classes.

It is mainly a time expression, not a literal spatial one.

So in this sentence, the idea is:

• once school is over, the children play on the playground behind the house

A Czech speaker would normally understand it that way unless the context suggests something more specific.