Pokaždé když hraji badminton, potřebuji badmintonovou raketu.

Yes. Pokaždé když means every time (that) or whenever.

It is a very common way to introduce a repeated situation:

• Pokaždé když prší... = every time it rains...
• Pokaždé když hraji badminton... = every time I play badminton...

In standard writing, you will very often see it written as Pokaždé, když... with a comma before když.

In standard Czech, yes: the usual punctuation is:

Pokaždé, když hraji badminton, potřebuji badmintonovou raketu.

Why? Because když hraji badminton is a subordinate clause, and Czech normally separates subordinate clauses with commas.

So the most standard version is:

• Pokaždé, když hraji badminton, potřebuji badmintonovou raketu.

The comma after badminton is definitely needed, and many teachers would also expect the comma after Pokaždé.

Yes, you can say both.

The verb is hrát = to play.
First person singular present can appear as:

• hraji — more standard, careful, or written
• hraju — very common in everyday spoken Czech

So both of these are normal:

• když hraji badminton
• když hraju badminton

Similarly, many speakers also say:

• potřebuji / potřebuju

The versions with -ji are often felt as a bit more standard; the versions with -ju are very common in speech.

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

Here:

• hraji = I play
• potřebuji = I need

So adding já is usually unnecessary.

You could say:

• Pokaždé, když já hraji badminton...

but that would usually sound marked or emphatic, as if you were contrasting yourself with someone else.

It actually is functioning as the direct object, so it is in the accusative. But the form looks the same.

The verb hrát takes the accusative:

• hrát fotbal
• hrát tenis
• hrát badminton

The noun badminton is masculine inanimate, and for many masculine inanimate nouns, the accusative singular is identical to the nominative singular. So:

• nominative: badminton
• accusative: badminton

That is why it looks unchanged.

Because potřebuji takes a direct object, and that object must be in the accusative case.

The dictionary form is:

• badmintonová raketa = a badminton racket

But in this sentence it becomes accusative singular:

• badmintonovou raketu

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

• nominative singular feminine: badmintonová raketa
• accusative singular feminine: badmintonovou raketu

So the endings tell you that this is the thing being needed.

The base form is potřebovat = to need.

Its present-tense first person singular form is:

• potřebuji = I need

This verb normally takes the accusative for the thing needed:

• Potřebuji čas. = I need time.
• Potřebuji pomoc. = I need help.
• Potřebuji badmintonovou raketu. = I need a badminton racket.

So badmintonovou raketu is in the accusative because it is the object of potřebuji.

Yes, it is normal and correct.

Czech often uses an adjective formed from the sport name:

• tenisová raketa = tennis racket
• badmintonová raketa = badminton racket

You may also hear structures like raketa na badminton, which is also understandable, but badmintonová raketa is a very natural standard expression.

Yes. Czech word order is more flexible than English word order.

The given sentence is perfectly natural, especially with the time clause first:

• Pokaždé, když hraji badminton, potřebuji badmintonovou raketu.

But you could also say:

• Potřebuji badmintonovou raketu pokaždé, když hraji badminton.

Both mean the same basic thing. The difference is mainly one of emphasis and information flow.

A rough guide:

• starting with Pokaždé, když... sets the scene first
• starting with Potřebuji... puts the main point first

Yes. In everyday speech, many Czechs would naturally say:

• Pokaždé, když hraju badminton, potřebuju badmintonovou raketu.

You might also hear:

• Vždycky, když hraju badminton, potřebuju badmintonovou raketu.

Notes:

• hraju is more colloquial than hraji
• potřebuju is more colloquial than potřebuji
• vždycky, když is a very common spoken alternative to pokaždé, když

So the original sentence is correct and clear, but a spoken version often uses more conversational forms.

The hardest part for many English speakers is ř.

A practical approximation:

• potřebuji sounds roughly like po-trzhe-boo-yi, but that is only an approximation.

About ř:

• It is a special Czech sound, something like a rolled or fricated r mixed with a zh-like quality.
• English does not have an exact equivalent.
• Learners are often understood if they pronounce it as something close to rž or even just a careful r at first.

Also note:

• -ji at the end is pronounced clearly, not reduced away.
• So hraji and potřebuji both end with a real -yi / -yee type ending.

If you cannot say ř perfectly yet, that is completely normal. It is one of the most difficult Czech sounds for learners.