Stomatolog daje anesteziju, tako da usta utrnu.

Yes. Stomatolog is a standard, medical-style word for dentist. In everyday speech, many people also say zubar.

So:

• stomatolog = dentist, more formal/medical
• zubar = dentist, more everyday/colloquial

In this sentence, stomatolog sounds perfectly natural.

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

So Stomatolog daje anesteziju can mean:

• The dentist gives anesthesia
• A dentist gives anesthesia
• even something more general like Dentists give anesthesia, depending on context

You figure out the exact meaning from the situation, not from an article.

Because anesteziju is the accusative singular form, used for the direct object of the verb.

• anestezija = nominative, dictionary form
• anesteziju = accusative

Since the dentist is giving/administering anesthesia, anestezija is the thing being given, so it takes the accusative:

• Stomatolog daje anesteziju.

This is a regular pattern for many feminine nouns ending in -a:

• voda → vodu
• knjiga → knjigu
• anestezija → anesteziju

Yes, daje literally means gives, but in this context English usually sounds more natural with administers or gives anesthesia.

So:

• daje = gives
• in medical English, often administers

Grammatically, daje is:

• 3rd person singular
• present tense

So it means he/she gives, or more generally the dentist gives/administers.

Here tako da means so, so that, or with the result that.

It introduces the result of the first action:

• Stomatolog daje anesteziju, tako da usta utrnu.
• The dentist administers anesthesia, so that the mouth goes numb.

A useful way to understand it is:

• first action: the dentist gives anesthesia
• result: the mouth goes numb

So this is about result, not just adding another action.

Because usta is a noun that is grammatically plural in Serbian, even when it refers to one person's mouth.

So although English says mouth (singular), Serbian uses usta, which behaves like a plural noun.

That is why other words in the sentence agree with it as plural.

This is one of those places where you should learn the noun as a fixed fact:

• usta = mouth
• but grammatically it acts as plural

Because its subject is usta, and usta is grammatically plural.

So:

• usta = plural subject
• utrnu = plural verb form

The verb utrnuti means to go numb, become numb, or lose feeling.

In this sentence, utrnu means:

• go numb
• become numb

So usta utrnu is literally something like:

• the mouth goes numb even though Serbian is using a grammatically plural noun.

Here it means go numb / become numb, not simply are numb.

That is an important nuance:

• utrnu focuses on the change
• English: become numb, go numb

So the sentence describes the effect of the anesthesia:

• the dentist administers it
• then the mouth becomes numb

If you wanted to emphasize the state is numb, Serbian would usually express that differently.

Yes. In Serbian, when tako da introduces a result clause, it is normally separated by a comma.

So this punctuation is standard:

• Stomatolog daje anesteziju, tako da usta utrnu.

The comma helps show that the second part explains the result of the first part.