Nach dem Zugausfall prüfe ich ruhig den Bearbeitungsstand.

Because nach in the sense of after takes the dative case.

• der Zugausfall = the train cancellation
• Dative singular masculine/neuter article = dem

So:

• nominative: der Zugausfall
• dative: nach dem Zugausfall

A learner might know den as the, but here den would be the accusative masculine form, which is not used after nach in this meaning.

Zugausfall is a compound noun:

• der Zug = train
• der Ausfall = cancellation, failure, breakdown

Together, der Zugausfall means train cancellation or the cancellation of a train.

German uses compound nouns very freely, so instead of a longer phrase like the cancellation of the train, it often builds one noun.

Because the subject is ich, and the verb must be conjugated.

The infinitive is prüfen = to check / to examine.

Present tense conjugation:

• ich prüfe
• du prüfst
• er/sie/es prüft
• wir prüfen
• ihr prüft
• sie/Sie prüfen

So ich prüfe means I check or I am checking, depending on context.

This is because German is a verb-second language in main clauses.

The sentence begins with the time phrase:

• Nach dem Zugausfall

When that element is placed first, the conjugated verb still has to stay in the second position, so the subject comes after it:

• Nach dem Zugausfall prüfe ich ...

Compare:

• Ich prüfe ruhig den Bearbeitungsstand.
• Nach dem Zugausfall prüfe ich ruhig den Bearbeitungsstand.

In English, starting with After the train cancellation does not change the normal subject-verb order much, but in German it triggers this inversion pattern.

Here ruhig is an adverb modifying prüfe.

Its basic meaning is calmly, quietly, or without stress. In this sentence it suggests that the speaker checks the status in a calm or composed way.

Depending on context, ruhig can also have other nuances in German, for example:

• Mach ruhig weiter. = Go ahead and continue.
• Bleib ruhig. = Stay calm.

But in your sentence, the most natural function is simply an adverb meaning something like calmly.

Because den Bearbeitungsstand is the direct object of prüfen, and Bearbeitungsstand is masculine.

The noun is:

• der Bearbeitungsstand

As a masculine noun, its accusative singular article becomes den:

• nominative: der Bearbeitungsstand
• accusative: den Bearbeitungsstand

So:

• Ich prüfe den Bearbeitungsstand. = I check the processing status / progress status

It is another compound noun:

• die Bearbeitung = processing, working on something, handling
• der Stand = state, status, current state

So der Bearbeitungsstand literally means something like processing status or current state of work.

Depending on context, it could refer to:

• the status of an application
• the progress of a task
• the current stage of editing or processing

German often packs this kind of idea into one noun.

Often yes, but not always exactly.

prüfen can mean:

• to check
• to examine
• to inspect
• to verify

It can sound a bit more formal or thorough than casual English check, depending on context.

For example:

• Ich prüfe den Bearbeitungsstand. = I check the status.
• Der Lehrer prüft die Schüler. = The teacher examines the students.
• Bitte prüfen Sie Ihre Angaben. = Please check your details.

So prüfen is a good verb here because it suggests looking at the status carefully.

Yes, that is grammatically possible, but the emphasis changes slightly.

• Nach dem Zugausfall prüfe ich ruhig den Bearbeitungsstand.

  • puts the time phrase first
  • emphasizes after the train cancellation

• Ich prüfe nach dem Zugausfall ruhig den Bearbeitungsstand.

  • starts with the subject
  • sounds a bit more neutral

German word order is flexible, but the element placed first usually gets more prominence.

Because German, like English, often uses the present tense for a current or general action that happens after some earlier event.

Here the cancellation happened first, and now the speaker says what they do afterward:

• Nach dem Zugausfall prüfe ich ...

This does not require a past tense verb in the main clause. It simply means that following that event, the speaker now checks the status.

So the timeline is:

1. the train is cancelled
2. after that, I check the status

That is completely normal in German.

Yes. It follows a very common German pattern:

1. time phrase first
2. conjugated verb second
3. subject next
4. other elements afterward

So the structure is:

• Nach dem Zugausfall = time expression
• prüfe = conjugated verb
• ich = subject
• ruhig = adverb
• den Bearbeitungsstand = object

This pattern is extremely common in German, so it is a good model sentence for learning word order.