Wir prüfen den Reifendruck mit der Luftpumpe.

Because Reifendruck is the direct object of the verb prüfen.

• The base noun is der Druck = pressure
• So der Reifendruck = tire pressure
• Der Reifendruck is masculine
• A masculine noun in the accusative case takes den

So:

• der Reifendruck = nominative
• den Reifendruck = accusative

Here, we are checking the tire pressure, so it is the thing being checked, which makes it the direct object.

Because the preposition mit always takes the dative case.

• die Luftpumpe = nominative
• der Luftpumpe = dative

So after mit, you must say:

• mit der Luftpumpe = with the air pump

This is a very common rule in German:

• mit
  • dative

Examples:

• mit dem Auto
• mit der Tasche
• mit den Freunden

Yes. In this sentence, prüfen means to check, to inspect, or to test.

Depending on context, prüfen can mean:

• to check
• to examine
• to test
• to inspect

Here, Wir prüfen den Reifendruck means We check the tire pressure.

It is slightly more formal or precise than some everyday English uses of check, but it is completely natural German.

Because in the wir form, many German verbs look exactly like the infinitive.

For prüfen:

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

So wir prüfen = we check

This is normal and not a special exception.

German often combines nouns into one word.

Reifendruck

• der Reifen = tire
• der Druck = pressure
• Reifendruck = tire pressure

Luftpumpe

• die Luft = air
• die Pumpe = pump
• Luftpumpe = air pump

A very important rule: the last part of the compound usually determines the gender.

So:

• der Druck → der Reifendruck
• die Pumpe → die Luftpumpe

That is why the sentence has:

• den Reifendruck (masculine accusative)
• der Luftpumpe (feminine dative after mit)

In this sentence, mit der Luftpumpe means using the air pump as the instrument/tool.

So the structure is:

• Wir prüfen = we check
• den Reifendruck = the tire pressure
• mit der Luftpumpe = with the air pump

In real life, this might mean checking the pressure using a pump that has a gauge, or using the pump in the process of checking. The phrase itself mainly tells you the means or tool used.

It does not by itself mean we inflate the tire.
If you wanted to say that, you would usually use a verb like aufpumpen.

Because German main clauses usually follow the V2 rule: the finite verb comes in the second position.

In this sentence:

• Wir = first position
• prüfen = second position

So: Wir prüfen den Reifendruck mit der Luftpumpe.

This is standard German word order.

Yes, but the finite verb still has to stay in second position in a normal main clause.

For example:

• Wir prüfen den Reifendruck mit der Luftpumpe.
• Mit der Luftpumpe prüfen wir den Reifendruck.
• Den Reifendruck prüfen wir mit der Luftpumpe.

All of these are grammatical, but the emphasis changes.

For example:

• Mit der Luftpumpe prüfen wir den Reifendruck. emphasizes with the air pump
• Den Reifendruck prüfen wir mit der Luftpumpe. emphasizes the tire pressure

English learners often expect a more fixed word order, but German is more flexible as long as the verb-second rule is respected.

Yes, that would also be natural.

Both prüfen and kontrollieren can mean to check.

Very roughly:

• prüfen often suggests examining or testing something
• kontrollieren often suggests checking whether something is correct or in order

In many everyday contexts, both work.
For tire pressure, prüfen is very common and sounds perfectly natural.

Here is the sentence broken down:

• Wir = subject = we
• prüfen = verb = check
• den Reifendruck = direct object in the accusative = the tire pressure
• mit der Luftpumpe = prepositional phrase = with the air pump

So the overall structure is:

Subject + verb + direct object + prepositional phrase

That makes it a very useful model sentence for German word order and case usage.