Nemoj piti kavu dok ti ne izvade krv.

Yes. Nemoj + infinitive is a very common way to make a negative command in Croatian.

• nemoj piti = don’t drink
• literally, nemoj comes from the verb moći in a fixed prohibitive form, but learners usually just treat nemoj / nemojte as the normal way to say don’t.

You will often see:

• nemoj — singular / informal
• nemojte — plural or polite

So here:

• Nemoj piti kavu = Don’t drink coffee

Because after nemoj, Croatian normally uses the infinitive.

So:

• nemoj piti = don’t drink not
• nemoj piješ

As for piti vs popiti:

• piti is imperfective: to drink, in a general/ongoing sense
• popiti is perfective: to drink up, to finish drinking

In this sentence, nemoj piti kavu sounds natural because the point is do not drink coffee at all before the blood draw, not specifically don’t finish a coffee.

Because kava is the direct object of piti, so it goes in the accusative case.

• nominative: kava = coffee
• accusative: kavu

So:

• piti kavu = to drink coffee

This is very common in Croatian:

• jesti kruh = to eat bread
• piti vodu = to drink water
• čitati knjigu = to read a book

Here dok means until.

So:

• dok ti ne izvade krv = until they take your blood

Be careful: in other contexts, dok can also mean while.

For example:

• Dok radim, slušam glazbu. = While I work, I listen to music.

But in your sentence, because of the structure with ne and the sense of waiting for something to happen, dok clearly means until.

This is one of the most important things to notice in Croatian.

After dok meaning until, Croatian very often uses ne even though the whole clause is not actually negative.

So:

• dok ti ne izvade krv literally looks like until they don’t take your blood
• but it really means until they take your blood

This ne is required by the Croatian structure and does not translate into English.

You can think of it as a grammatical marker that often appears after words like dok or dok ne in until clauses.

Izvade is 3rd person plural: they take out / they remove.

Here it refers to unnamed people such as:

• the nurse
• the doctor
• the medical staff

Croatian often uses 3rd person plural when the doer is unspecified, much like English sometimes says:

• They took my blood even if we do not mean a specific group already mentioned.

So:

• dok ti ne izvade krv = until they take your blood

Ti here is the dative pronoun meaning to you / from you, depending on the verb and context.

In this sentence:

• izvade ti krv

literally means something like:

• they take blood from you

Croatian often uses a dative pronoun for the person affected by the action.

So:

• ti = from you / for you / to you, depending on the situation
• here the natural English meaning is from you

That is why the full phrase means:

• until they take your blood or more naturally,
• until they draw your blood

Here krv is the direct object of izvade, and for this noun the accusative singular is the same as the nominative singular.

So:

• nominative: krv = blood
• accusative: krv

That is why you see:

• izvaditi krv = to take/draw blood

Not every noun changes visibly in the accusative. Some do, some do not.

Literally, izvaditi krv means to take out blood or to remove blood.

In medical English, the most natural translation is:

• draw blood
• take blood

So the sentence is not about some dramatic loss of blood; it is just the normal medical expression for a blood test or blood draw.

Yes. A few alternatives are possible, depending on style and nuance.

For example:

• Nemoj piti kavu prije vađenja krvi.
  = Don’t drink coffee before the blood draw.

• Nemoj piti kavu dok ti ne uzmu krv.
  = Don’t drink coffee until they take your blood.

• Nemoj popiti kavu prije nego ti izvade krv.
  = Don’t drink a coffee before they draw your blood.

Your original sentence is very natural and idiomatic.

It breaks down like this:

• Nemoj = don’t
• piti = drink
• kavu = coffee
• dok = until
• ti = from you / to you
• ne izvade = they take out
• krv = blood

So the pattern is:

• Nemoj + infinitive = negative command
• followed by
• a dok ... ne ... clause = until ...

In a more literal word-for-word gloss:

• Nemoj piti kavu dok ti ne izvade krv.
• Don’t drink coffee until they take blood from you.

Natural English:

• Don’t drink coffee until they draw your blood.