Skupina hraje deskovou hru.

Because Czech verb agreement follows the grammatical form of the subject, not the idea behind it.

• skupina = group
• It is a singular noun in Czech
• So the verb is also singular: hraje = plays / is playing

This is similar to English The group is playing, not The group are playing in standard American English.

Hraje is the 3rd person singular present tense form of hrát (to play).

So:

• já hraju / hraji = I play / am playing
• ty hraješ = you play / are playing
• on/ona/ono hraje = he/she/it plays / is playing

In this sentence, skupina is treated like it, so Czech uses hraje.

It can mean either one, depending on context.

Czech present tense often covers both:

• plays
• is playing

So Skupina hraje deskovou hru can mean:

• The group plays a board game
• The group is playing a board game

If you need to know which one is meant, the wider context usually makes it clear.

Because hru is the accusative singular form of hra.

• Dictionary form: hra = game
• In this sentence, game is the direct object of plays
• Direct objects usually go in the accusative case
• So hra changes to hru

This is a very common pattern for feminine nouns ending in -a:

• žena → ženu
• kniha → knihu
• hra → hru

Because adjectives in Czech must agree with the noun they describe.

The noun hru is:

• feminine
• singular
• accusative

So the adjective must also be:

• feminine
• singular
• accusative

That gives:

• dictionary form: desková = board
• in this sentence: deskovou

So:

• desková hra = a board game / board game
• hraje deskovou hru = plays a board game

It is a noun phrase in the accusative singular feminine:

• deskovou = adjective, accusative singular feminine
• hru = noun, accusative singular feminine

Together they mean board game as the object of the verb.

The nominative form is:

• desková hra

But after hrát here, it becomes:

• deskovou hru

Because Czech has no articles.

English says:

• a group
• the group
• a board game
• the board game

Czech usually just says:

• skupina
• desková hra / deskovou hru

Whether the meaning is a or the depends on context.

So Skupina hraje deskovou hru could mean:

• The group is playing a board game
• A group is playing a board game
• sometimes even The group is playing the board game, if context already identifies which one

The basic neutral word order here is:

• Skupina hraje deskovou hru.

This is the most straightforward way to say it.

But Czech word order is more flexible than English, because the case endings show what role words have. So other orders are possible, for example:

• Deskovou hru hraje skupina.
• Hraje skupina deskovou hru.

These alternatives usually change the emphasis or sound more marked. For a learner, the original version is the best neutral pattern.

Yes. Then it simply means The group is playing a game.

• hraje hru = is playing a game
• hraje deskovou hru = is playing a board game

So deskovou just adds more information about what kind of game it is.

A rough English-friendly pronunciation is:

SKOO-pi-na HRA-ye DES-ko-voh HROO

A few useful notes:

• skupina = stress on the first syllable: SKU-
• hraje: the h is pronounced; je sounds roughly like ye
• deskovou: the ending -ou is a diphthong, roughly like a long oh/oo glide
• hru: the r is trilled or tapped, and u sounds like oo

Also remember: Czech stress is almost always on the first syllable of the word.

Often yes, but not always. It depends on what is being played.

Examples:

• hrát deskovou hru = play a board game
• hrát fotbal = play football
• hrát šachy = play chess

But with musical instruments, Czech often uses na:

• hrát na kytaru = play the guitar
• hrát na piano = play the piano

So in your sentence, hrát takes a direct object in the accusative because deskovou hru is the thing being played.

Yes: although it refers to multiple people, it is grammatically just a normal feminine singular noun.

That affects several things:

• verb: skupina hraje
• adjective with it: velká skupina = a big group
• pronoun referring to it would also be singular

So Czech treats skupina as one unit grammatically, even if the meaning involves many people.