Nach der Operation atmet der Arzt auf.

Because nach is a preposition that takes the dative case when it means after in a time sense.

• base form: die Operation
• dative singular: der Operation

So:

• nach der Operation = after the operation

This is a very common pattern in German:

• nach dem Essen = after the meal
• nach der Arbeit = after work
• nach dem Film = after the film

Here, nach means after in a temporal sense.

So Nach der Operation means After the operation or Following the operation.

Be careful: nach can also have other meanings in German, such as:

• to / toward with cities or countries without an article
  
  • nach Berlin
• according to
  
  • nach dem Arzt = according to the doctor

But in this sentence, it clearly means after.

Because the verb is aufatmen, which is a separable verb.

The full verb is:

• aufatmen = to breathe a sigh of relief / to feel relieved

In a normal main clause, separable verbs split:

• the conjugated part goes in the usual verb position
• the prefix goes to the end

So:

• infinitive: aufatmen
• main clause: Der Arzt atmet auf.

That is why you get:

• Nach der Operation atmet der Arzt auf.

Not really. Even though it is made of atmen + auf, you should learn aufatmen as one unit.

In this sentence, it is usually understood idiomatically as:

• to breathe a sigh of relief
• to feel relieved

So the doctor is not simply breathing in a physical sense; the sentence suggests relief after the operation.

In German main clauses, the finite verb must be in second position. This is called the V2 rule.

Here, the sentence begins with a time expression:

• Nach der Operation

Since that whole phrase takes the first position, the verb must come next:

• Nach der Operation | atmet | der Arzt | auf.

So the subject der Arzt comes after the verb.

You could also say:

• Der Arzt atmet nach der Operation auf.

That version starts with the subject, but both are grammatical.

Der Arzt is in the nominative case because it is the subject of the sentence—the person doing the action.

The verb is atmet auf, and the doctor is the one who is relieved.

So:

• der Arzt = nominative singular masculine

If it were an object, the form might change, for example:

• nominative: der Arzt
• accusative: den Arzt
• dative: dem Arzt

But here it stays der Arzt because it is the subject.

Here it is intransitive, which means it does not take a direct object.

You do not say something like:

• Er atmet die Erleichterung auf. ✘

Instead, you simply say:

• Er atmet auf. = He breathes a sigh of relief.

If you want to explain the reason for the relief, you usually add a phrase such as:

• nach der Operation
• vor Erleichterung
• als alles vorbei ist

Yes. German word order is flexible, as long as the main-clause verb rule is respected.

These are possible:

• Nach der Operation atmet der Arzt auf.
• Der Arzt atmet nach der Operation auf.

Both are correct.

The difference is mainly one of emphasis:

• Nach der Operation ... emphasizes the time first
• Der Arzt ... emphasizes the subject first

The separable prefix auf still stays at the end in a main clause.

Yes. In German, all nouns are capitalized, not just proper names.

So:

• die Operation
• der Arzt

This is one of the first big spelling differences English speakers notice.

Usually yes, or at least to one that is understood from context.

Because German uses the definite article here:

• der Operation

it sounds like the operation rather than an operation.

In real context, that could mean:

• an operation already mentioned
• the operation the doctor has just completed
• the relevant surgery everyone knows about

Yes. It behaves like a normal separable verb.

Examples:

• Der Arzt atmet auf. = The doctor breathes a sigh of relief.
• Der Arzt atmete auf. = The doctor breathed a sigh of relief.
• Der Arzt hat aufgeatmet. = The doctor has breathed a sigh of relief.

Notice the past participle:

• aufgeatmet

The prefix auf- rejoins the verb in the participle.

In a subordinate clause, the separable verb is not split in the same way. The full verb form goes to the end.

For example:

• ..., weil der Arzt nach der Operation aufatmet.
• ..., weil der Arzt nach der Operation aufgeatmet hat.

So compare:

• main clause: Der Arzt atmet auf.
• subordinate clause: ..., weil der Arzt aufatmet.

That is a very common contrast in German separable verbs.