Nach der Passkontrolle gehen wir zum Zoll.

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

• die Passkontrolle = nominative/accusative
• der Passkontrolle = dative

Since Passkontrolle is a feminine noun (die Passkontrolle), its dative singular form is der Passkontrolle.

So:

• nach der Passkontrolle = after passport control

No. Nach has more than one common meaning.

In this sentence, nach means after because it is talking about time:

• nach der Passkontrolle = after passport control

But nach can also mean to, especially with cities, countries without articles, and directions:

• nach Berlin = to Berlin
• nach Deutschland = to Germany
• nach links = to the left

So you have to tell from the context which meaning it has.

Because German very often forms compound nouns by joining words together.

Here:

• Pass = passport
• Kontrolle = control/check

So Passkontrolle literally combines them into one noun: passport control.

A useful rule: The last part of a German compound usually determines the gender and basic grammatical behavior.

Since Kontrolle is feminine (die Kontrolle), Passkontrolle is also feminine (die Passkontrolle).

Because German main clauses follow the verb-second rule.

The first position in the sentence is occupied by Nach der Passkontrolle. Once that first slot is taken, the conjugated verb must come next:

• Nach der Passkontrolle | gehen | wir | zum Zoll

So gehen is in second position, and wir comes after it.

This is completely normal in German.

Yes. That is also correct.

Both are grammatical:

• Nach der Passkontrolle gehen wir zum Zoll.
• Wir gehen nach der Passkontrolle zum Zoll.

The difference is mainly emphasis:

• Nach der Passkontrolle ... puts the time sequence first
• Wir gehen ... starts more neutrally with the subject

English learners often expect the subject to come first, but German allows other elements in first position very naturally.

Zum is a contraction of zu dem.

• zu = to
• dem = dative form of der

So:

• zu dem Zoll → zum Zoll

German uses this contraction very often, and in normal speech and writing zum is the usual form.

Zu takes the dative case.

That is why der Zoll becomes dem Zoll after zu:

• der Zoll = nominative
• zu dem Zoll = dative
• contracted: zum Zoll

So both prepositions in this sentence take dative:

• nach der Passkontrolle
• zum Zoll

Because Zoll is masculine:

• der Zoll

So after zu, it becomes:

• zu dem Zoll → zum Zoll

You would use zur only with a feminine noun:

• zu der Bank → zur Bank

So:

• masculine/neuter: zum
• feminine: zur

It can be understood as both, and that is normal.

In this kind of travel sentence, der Zoll usually refers to the customs area / customs checkpoint / customs authorities in a practical sense. German often uses the short noun Zoll where English might say customs.

So zum Zoll gehen is the natural way to say go to customs.

Yes. German often uses the present tense for future meaning when the context already makes the time clear.

Here, the sequence after passport control already tells you what happens next, so gehen wir can naturally mean:

• we go
• we are going
• we will go

depending on context.

German does not need a separate future form as often as English does. A form with werden is possible in some contexts, but here the simple present is the most natural choice.