Lekar će proveriti nivo gvožđa i hemoglobin.

• Lekar = doctor, the subject, in the nominative case
• će proveriti = will check, the future tense
• nivo gvožđa = the level of iron / iron level, the first object
• i = and
• hemoglobin = hemoglobin, a second object

So structurally, the sentence is basically:

[Subject] + [future verb] + [object 1] + [and] + [object 2]

Serbian commonly forms the future with a present-tense form of hteti plus the infinitive.

Here:

• će = will
• proveriti = to check

So:

• Lekar će proveriti = The doctor will check

This is a very standard future construction in Serbian.

Because after nivo Serbian usually uses the genitive case to mean level of something.

• gvožđe = iron
• gvožđa = of iron

So:

• nivo gvožđa = level of iron / iron level

This is a very common pattern:

• nivo šećera = blood sugar level / sugar level
• nivo holesterola = cholesterol level

Because hemoglobin here is functioning as a direct object, and for a masculine inanimate noun in the singular, the accusative is usually the same as the nominative.

So:

• nominative: hemoglobin
• accusative: hemoglobin

That is why the form stays the same.

Yes, depending on what exactly you want to say.

As written, nivo gvožđa i hemoglobin most naturally means:

• the iron level and hemoglobin

So the doctor is checking two things:

1. nivo gvožđa
2. hemoglobin

If you want hemoglobin to depend on nivo too, then you would say:

• nivo gvožđa i hemoglobina

That means:

• the level of iron and hemoglobin
• more naturally in English, iron and hemoglobin levels

So the given sentence is understandable, but hemoglobina would be used if both nouns are governed by nivo.

Because Serbian has no articles.

English says:

• the doctor
• a doctor

Serbian just says:

• lekar

Whether it means a doctor or the doctor depends on context.

So Lekar će proveriti... can mean either:

• The doctor will check...
• A doctor will check...

In most real situations, context makes it clear.

Because proveriti is perfective, and perfective verbs are often used when talking about a single completed action.

So:

• proveriti = to check once, to complete the check
• proveravati = to be checking / to check repeatedly or habitually

In this sentence, the meaning is a single future action:

• The doctor will check...

That is why proveriti fits well.

If you said:

• Lekar će proveravati... it would suggest repeated or ongoing checking.

Yes. Both are common and correct.

You can say:

• Lekar će proveriti nivo gvožđa i hemoglobin.
• Lekar će da proveri nivo gvožđa i hemoglobin.

Both mean the same thing: The doctor will check...

The version with infinitive (će proveriti) is very standard and often feels a bit more formal or neutral in writing.
The version with da + present (će da proveri) is also very common in speech.

No. Lekar and doktor are both common.

• lekar = doctor, physician
• doktor = doctor

In many situations, both work:

• Lekar će proveriti...
• Doktor će proveriti...

Lekar can sound a bit more specifically like physician/medical doctor, while doktor is also extremely common in everyday speech.

Grammatically, lekar is a masculine noun, but it can sometimes be used generically for the profession. If you want to specify a female doctor, many speakers use:

• lekarka
• or doktorka

So you may hear:

• Lekarka će proveriti nivo gvožđa i hemoglobin.

That specifically means:

• The female doctor will check the iron level and hemoglobin.

Not completely. Serbian word order is more flexible than English word order, although some orders sound more neutral than others.

The given sentence:

• Lekar će proveriti nivo gvožđa i hemoglobin.

is a neutral, natural order.

You can change things for emphasis, but then the sentence may sound more marked. One important detail is that će is a clitic, so it usually appears near the beginning of the clause, typically in second position.

So for learners, the safest neutral pattern is:

• Lekar će proveriti...

That is the one you should imitate first.