Nach der Reise räumen wir den Kofferraum aus.

Because nach takes the dative case here.

The basic noun is die Reise (feminine nominative). After nach in the sense of after, it changes to dative singular:

• nominative: die Reise
• dative: der Reise

So Nach der Reise means after the trip/journey.

Here, nach means after.

That can be confusing, because nach also often means to with places, especially towns and countries without an article:

• nach Berlin = to Berlin
• nach Deutschland = to Germany

But in Nach der Reise, it is a time expression, so it means after the trip.

Because the verb is ausräumen, which is a separable-prefix verb.

Its parts are:

• aus- = prefix
• räumen = verb stem

In a normal main clause, German separates them:

• Wir räumen den Kofferraum aus.

So the finite verb räumen goes to the usual verb position, and the prefix aus moves to the end.

You will see the full verb together in places like these:

• infinitive: den Kofferraum ausräumen
• subordinate clause: ..., weil wir den Kofferraum ausräumen

Because in a German main clause, the finite verb must be in second position.

German allows different elements to come first. If you put Nach der Reise first, then the verb must come immediately after it:

• Nach der Reise räumen wir den Kofferraum aus.

So the order is:

1. Nach der Reise
2. räumen
3. wir
4. the rest of the sentence

If the subject comes first, that also works:

• Wir räumen nach der Reise den Kofferraum aus.

Both are correct. The original sentence just puts a little more emphasis on after the trip.

Because den Kofferraum is the direct object of the verb ausräumen, so it is in the accusative case.

The noun is:

• nominative: der Kofferraum
• accusative: den Kofferraum

This happens because Kofferraum is masculine. For masculine nouns, the article changes from der to den in the accusative.

The dictionary form is ausräumen.

That is the infinitive. In a sentence like this one, it splits because it is separable:

• infinitive: ausräumen
• present tense main clause: räumen ... aus

So if you look it up in a dictionary, look for ausräumen, not just räumen.

Yes. That is also correct.

German is flexible about word order, as long as the verb placement rules are followed. These are both fine:

• Nach der Reise räumen wir den Kofferraum aus.
• Wir räumen nach der Reise den Kofferraum aus.

The difference is mainly focus:

• Nach der Reise ... puts the time expression in the spotlight.
• Wir ... starts more neutrally with the subject.

Because German commonly forms compound nouns by writing them as a single word.

Kofferraum is made from:

• Koffer = suitcase
• Raum = room/space

So literally it is something like suitcase space, which in normal English is trunk or boot.

This is very typical in German:

• Bahnhof = train + yard/station
• Handschuh = hand + shoe = glove

Yes. It is a prepositional phrase that gives a time reference: after the trip.

It is not a full sentence by itself, but it functions as one complete element at the start of the sentence. That is why it can take the first position before the verb:

• Nach der Reise | räumen | wir | den Kofferraum aus

In German grammar, that whole phrase counts as the first element of the clause.