Сімейний лікар написав направлення на аналіз крові.

Yes. Сімейний лікар literally means family doctor, and in practice it is the usual Ukrainian term for a general practitioner / primary care doctor.

• сімейний = family-related, family
• лікар = doctor

So this is the doctor you normally register with for general medical care.

Because сімейний agrees with лікар in:

• gender: masculine
• number: singular
• case: nominative

Here сімейний лікар is the subject of the sentence, so it is in the nominative case.

If the form changed, the adjective would change too. For example:

• сімейного лікаря = the family doctor (accusative/genitive form)
• сімейним лікарем = by/with the family doctor (instrumental)

Написав is the past tense, masculine singular form of написати = to write.

It matches the subject лікар, which is grammatically masculine singular.

So:

• лікар написав = the doctor wrote
• лікарка написала = the female doctor wrote

In Ukrainian past tense, the verb often shows the gender of the subject in the singular.

This is about aspect.

• писати / писав = imperfective, focuses on the process: was writing / wrote habitually
• написати / написав = perfective, focuses on the completed result: wrote / has written

In this sentence, the idea is that the doctor completed the referral, so написав is the natural choice.

Направлення means a referral or medical referral form/order.

In a medical context, it is the document a doctor gives you so you can have a test, visit a specialist, or receive some procedure.

Grammatically, направлення is a neuter singular noun.

Here на means something like for.

So:

• направлення на аналіз крові = a referral for a blood test

This pattern is very common in Ukrainian:

• квиток на потяг = a ticket for the train
• запис на прийом = an appointment booking
• направлення на обстеження = a referral for an examination

After на in this meaning, Ukrainian usually uses the accusative case.

Аналіз is in the accusative singular, but for many inanimate masculine nouns, the accusative has the same form as the nominative.

So:

• nominative: аналіз
• accusative: аналіз

That is why after на you still see аналіз.

Because крові is the genitive singular form of кров.

Ukrainian often uses the genitive after a noun to mean of ...:

• аналіз крові = analysis of blood
• склянка води = glass of water
• чашка чаю = cup of tea

So аналіз крові literally means analysis of blood, which English usually expresses as blood test.

Yes, it is very common and natural.

Literally it means analysis of blood, but in normal English we usually say blood test. So the Ukrainian wording is slightly more literal than the English one.

You may also hear related phrases like:

• здати аналіз крові = to have a blood test / to give blood for testing
• результати аналізу крові = blood test results

Yes. Ukrainian word order is more flexible than English because case endings show grammatical relationships.

The neutral order here is:

• Сімейний лікар написав направлення на аналіз крові.

But other orders are possible for emphasis, for example:

• Направлення на аналіз крові написав сімейний лікар.
  Emphasis on who wrote it.
• На аналіз крові сімейний лікар написав направлення.
  Emphasis on what the referral was for.

Even though the order changes, the basic meaning stays the same.

Yes. Лікар is the general Ukrainian word for a medical doctor.

Depending on context, it can be translated as:

• doctor
• physician
• sometimes more specifically medical doctor

If you want to emphasize that the doctor is female, Ukrainian often uses:

• лікарка = female doctor

So you might also hear:

• Сімейна лікарка написала направлення...

There are other natural options too.

Besides написав направлення, Ukrainians also often say:

• виписав направлення = issued a referral
• дав направлення = gave a referral

The sentence you have is completely understandable and natural, but in medical settings виписав направлення is especially common.

The main change would be in the noun and the past-tense verb:

• Сімейна лікарка написала направлення на аналіз крові.

Changes:

• сімейний → сімейна to match the feminine noun
• лікар → лікарка
• написав → написала

So adjective, noun, and past-tense verb all agree with the feminine subject.