Lekar će mi dati recept.

Serbian does not have articles like English the and a/an.

So:

• lekar can mean doctor, a doctor, or the doctor
• recept can mean prescription, a prescription, or the prescription

Context tells you which one is meant. If Serbian wants to be more specific, it can use words like taj (that) or ovaj (this).

A quick breakdown:

• Lekar = doctor
• će = future auxiliary, roughly will
• mi = to me
• dati = to give
• recept = prescription

Literally, the structure is close to:

Doctor will to-me give prescription.

Natural English: The doctor will give me a prescription.

This is the future tense.

It is built with:

• the auxiliary će
• plus the infinitive dati

So:

• ću dati = I will give
• ćeš dati = you will give
• će dati = he/she/it will give

In this sentence, the subject is lekar, which is third person singular, so the auxiliary is će.

Because će is a clitic: a short unstressed word that normally cannot stand at the beginning of a sentence.

So after a word like Lekar, you get:

• Lekar će mi dati recept.

But if the future auxiliary would come first, it usually attaches to the verb:

• Daće mi recept. = He/she will give me a prescription.

So both patterns are normal:

• Lekar će mi dati recept.
• Daće mi recept.

Yes. That is also a common Serbian future construction, especially in speech.

Compare:

• Lekar će mi dati recept.
• Lekar će da mi da recept.

Both mean the same thing.

Very roughly:

• će + infinitive = often felt as a bit more neutral or written-style
• će da + present = very common in everyday speech

So da mi da may look strange to an English speaker, but it is completely normal Serbian.

Mi is the short dative form of ja (I).

With dati (to give), the person receiving something is put in the dative:

• dati nekome nešto = to give someone something

So here:

• mi = to me

That is why Serbian does not use me/mene here. Those are accusative forms, not dative.

Compare:

• dao mi je recept = he gave me a prescription
• video me je = he saw me

Different verbs, different cases.

Normally, no. Lekar će dati mi recept sounds unnatural/wrong.

The reason is that mi is also a clitic, and Serbian clitics tend to appear near the beginning of the sentence, in a special position.

So the natural order is:

• Lekar će mi dati recept.

Not:

• Lekar će dati mi recept.

Short forms like će, mi, ti, ga, je, se follow strict placement rules much more than English words do.

Recept is the direct object, so it is in the accusative case.

But for many masculine inanimate nouns in Serbian, the accusative singular looks exactly the same as the nominative singular.

So:

• recept (nominative)
• recept (accusative)

That is normal.

Compare that with a masculine animate noun:

• lekar (nominative)
• lekara (accusative)

For example:

• Vidim lekara. = I see the doctor.

So recept stays the same because it is masculine inanimate.

This is about aspect, which is very important in Serbian.

• dati = perfective
  one complete act of giving
• davati = imperfective
  repeated giving, habitual giving, or ongoing process

In this sentence, the doctor is expected to perform one complete action: give you a prescription once. So dati is the natural choice.

• Lekar će mi dati recept. = one completed future act

If you said:

• Lekar će mi davati recepte.

that would suggest repeated or habitual giving of prescriptions.

Yes, Serbian word order is more flexible than English word order.

The sentence:

• Lekar će mi dati recept.

is a neutral, natural version.

But you can change the order for emphasis, for example:

• Recept će mi dati lekar.
  Emphasis on the doctor as the one who will give it

• Meni će lekar dati recept.
  Emphasis on to me

However, the clitics still have to follow their placement rules, so not every rearrangement is possible.

Yes, if the context already makes it clear who is being talked about.

• Daće mi recept. = He/She will give me a prescription.

Serbian often leaves out subjects when they are understood from context.

In your full sentence, lekar is included because the speaker wants to name the subject explicitly.

People really say this. It is completely normal Serbian.

It sounds natural and standard. In everyday speech, you may also hear:

• Doktor će mi dati recept.
• Lekar će da mi da recept.

But your original sentence is perfectly idiomatic and something a native speaker would absolutely say.