Zamíchali jsme balíček, dali žetony na stůl a desková hra začala.

Because Czech forms the past tense with:

• a past participle-like form: zamíchali
• plus an auxiliary from být: jsme

So zamíchali jsme means we shuffled / we mixed.

For we, the auxiliary is jsme. Compare:

• zamíchal jsem = I shuffled
• zamíchal jsi = you shuffled
• zamíchali jsme = we shuffled

In modern Czech, this is the normal way to make the past tense.

Because Czech usually drops subject pronouns when they are not needed.

The form jsme already tells you the subject is we, so my is unnecessary unless you want emphasis or contrast.

• Zamíchali jsme balíček = We shuffled the deck
• My jsme zamíchali balíček = We shuffled the deck (emphatic, maybe contrasting with someone else)

English needs we, but Czech often does not.

Jsme is a clitic, and clitics in Czech usually go in the second position of the clause.

So in Zamíchali jsme balíček, the first element is Zamíchali, and jsme comes right after it.

That is why this order sounds natural:

• Zamíchali jsme balíček

Other word orders are possible for emphasis, for example:

• Balíček jsme zamíchali

But starting a clause with jsme is generally not the normal neutral order.

They are direct objects, so they are in the accusative case. But for these nouns, the accusative happens to look the same as the nominative.

• balíček is masculine inanimate singular
• žetony is masculine inanimate plural

For many masculine inanimate nouns:

• nominative singular = accusative singular
• nominative plural = accusative plural

So:

• balíček = nominative or accusative singular
• žetony = nominative or accusative plural

That is normal. The case is determined by function, not only by visible ending.

Because na can take different cases depending on meaning:

• na + accusative = movement onto something
• na + locative = location on something

Here the chips are being put onto the table, so there is movement:

• dali žetony na stůl = put the chips onto the table

If you were just describing where they are, you would use the locative:

• Žetony jsou na stole = The chips are on the table

This is a very common Czech pattern.

Because the subject hra is feminine singular, and in the past tense Czech agrees with the subject in gender and number.

So:

• hra začala = the game started
• film začal = the film started
• auto začalo = the car started
• hry začaly = the games started

Since desková hra is feminine singular, the correct form is začala.

Czech often uses an adjective + noun structure where English uses a noun + noun structure.

So:

• board game in English
• desková hra in Czech

The adjective desková comes from deska and means something like board-type / relating to a board.

This is the normal Czech expression for board game.

They are perfective:

• zamíchali ← zamíchat
• dali ← dát
• začala ← začít

Perfective verbs present actions as completed events. That fits this sentence well, because it describes a sequence of finished steps in a game:

1. the deck was shuffled
2. the chips were put on the table
3. the game started

Using perfective verbs makes the sentence sound like a clear narrative progression.

Because the sentence contains coordinated clauses:

• Zamíchali jsme balíček
• dali žetony na stůl
• a desková hra začala

The first two clauses are simply placed next to each other, so a comma is used between them. The last clause is introduced by a, and in this kind of simple coordination Czech normally does not put a comma before a.

So the punctuation is natural and correct.

Yes. Czech word order is flexible, so other versions are possible depending on emphasis.

For example:

• Balíček jsme zamíchali, žetony dali na stůl a desková hra začala.
• Zamíchali jsme balíček, na stůl dali žetony a desková hra začala.

These are all understandable, but the original sentence sounds natural and neutral as a simple narration of actions.

Czech does not always need an extra word like then.

The sequence is already clear because:

• the clauses are listed one after another
• the verbs are perfective
• the conjunction a links the events naturally

So Zamíchali jsme balíček, dali žetony na stůl a desková hra začala naturally reads as:

first this happened, then this, then this.

English often uses extra linking words more freely, but Czech can leave the sequence implicit.