Hrajeme badminton na badmintonovém kurtu.

Hrajeme is the 1st person plural present tense form of hrát.

So:

• hrát = to play
• hrajeme = we play / we are playing

In Czech, the present tense often covers both:

• We play badminton (habitually)
• We are playing badminton (right now)

The exact meaning usually comes from context.

Czech often leaves out subject pronouns when they are already clear from the verb ending.

In hrajeme, the ending -eme already tells you it means we.

So:

• Hrajeme badminton. = We play badminton.
• My hrajeme badminton. also means We play badminton, but my adds emphasis, like we in we are the ones playing badminton.

This is very normal in Czech.

It actually is in the correct case here: the accusative, because it is the direct object of hrajeme.

With hrát + a sport/game, Czech uses the direct object:

• hrát badminton
• hrát fotbal
• hrát tenis

The reason badminton looks unchanged is that many masculine inanimate nouns have the same form in the nominative and accusative singular.

So even though the case changes grammatically, the word form may not visibly change.

After na when it means in/on/at a place (location, not motion), Czech uses the locative case.

So:

• basic form: badmintonový kurt
• after na for location: na badmintonovém kurtu

Both words change because the adjective must agree with the noun:

• badmintonový → badmintonovém
• kurt → kurtu

This whole phrase means on/at a badminton court.

Because Czech uses different prepositions from English, and they do not match one-to-one.

For sports courts and similar surfaces/venues, Czech normally says:

• na kurtu = on/at the court
• na hřišti = on the field / on the pitch
• na stadionu = at the stadium

Using v here would sound unnatural. So even if English might say on or at, Czech idiomatically uses na.

This is a very important Czech pattern:

• na + locative = location
• na + accusative = motion toward a place

So:

• Jsme na kurtu. = We are on/at the court.
• Jdeme na kurt. = We are going to the court.

In your sentence, na badmintonovém kurtu is locative because it describes where the playing happens.

It comes from the noun badminton and the adjective suffix -ový.

So:

• badminton → badmintonový = badminton-related

Then it changes form to match kurtu in case, gender, and number:

• badmintonový kurt = a badminton court
• na badmintonovém kurtu = on/at a badminton court

This kind of adjective is very common in Czech:

• tenisový kurt = tennis court
• fotbalový míč = football/soccer ball
• hokejový zápas = hockey match

No, Czech word order is fairly flexible.

Your sentence:

• Hrajeme badminton na badmintonovém kurtu.

is perfectly natural. But other orders are also possible, depending on emphasis:

• Na badmintonovém kurtu hrajeme badminton.
  Emphasizes the location.
• Badminton hrajeme na badmintonovém kurtu.
  Emphasizes the sport.

Even though the word order can move, the cases still show the grammatical relationships.

Yes, often you can.

For example:

• Hrajeme badminton na kurtu.

This is very natural if the context already makes it clear what kind of court you mean.

The full sentence with both badminton words is grammatically fine, but it sounds a bit more explicit than everyday speech usually needs. Learners often see this kind of full wording because it makes the structure very clear.

Normally, no. For sports and games like badminton, you use hrát:

• hrajeme badminton
• hraju tenis
• hrají šachy

Hrát si usually means to play around, especially in the sense of children playing, playing with toys, or amusing oneself.

So:

• Děti si hrají. = The children are playing.
• Hrajeme badminton. = We are playing badminton.

For this sentence, hrát is the correct verb.

A rough English-friendly guide would be:

HRA-ye-me BAD-min-ton na bad-min-TO-no-vyem KOOR-too

A few useful notes:

• hrajeme: the hr cluster is real in Czech; the j sounds like English y
• badmintonovém: the stress is on the first syllable, as in Czech words generally
• kurtu: u is pronounced like oo

Czech stress is usually on the first syllable of each word:

• HRAjeme
• BADminton
• NA
• BADmintonovém
• KURtu