Bevor wir das Regal montieren, müssen wir erst das Zimmer lüften.

Because bevor introduces a subordinate clause. In German, the conjugated verb usually goes to the end of a subordinate clause.

So:

• Bevor = before
• wir das Regal montieren = we assemble the shelf

The structure is:

• Bevor + subject + other elements + verb

That is why you get:

• Bevor wir das Regal montieren not
• Bevor wir montieren das Regal

Because the sentence begins with the subordinate clause Bevor wir das Regal montieren. When a subordinate clause comes first, it takes up the first position in the main clause structure, so the conjugated verb must come immediately after it.

German main clauses follow the verb-second rule.

So the pattern is:

• [First element] + conjugated verb + subject + ...

Here, the first element is the whole clause:

• Bevor wir das Regal montieren, müssen wir erst das Zimmer lüften.

If you put the main clause first, you get normal word order:

• Wir müssen erst das Zimmer lüften, bevor wir das Regal montieren.

Here erst means something like first or before anything else.

So:

• müssen wir erst das Zimmer lüften
  = we first have to air out the room / we have to air out the room first

It emphasizes the order of actions:

1. air out the room
2. assemble the shelf

In everyday German, erst is very common in this kind of sentence.

They are similar, but not always identical.

In this sentence, erst works very naturally and means first / before that.
You could also say:

• Bevor wir das Regal montieren, müssen wir zuerst das Zimmer lüften.

That is also correct.

A rough difference:

• zuerst often clearly means first, as the first step
• erst can also carry a sense of only then / before that happens

In this sentence, both are possible, but erst sounds very natural.

Because both nouns are neuter in German:

• das Regal = the shelf / bookcase
• das Zimmer = the room

They are also in the accusative case here, because they are the direct objects of the verbs:

• das Regal montieren = assemble the shelf
• das Zimmer lüften = air out the room

For neuter singular, the article stays das in both nominative and accusative, so there is no visible change.

Montieren means to assemble, to put together, or sometimes to install, depending on context.

In this sentence, das Regal montieren most likely means:

• to assemble the shelf
• to put the shelf together

If this is furniture from a store, montieren is a very natural verb.

Here lüften means to air out or to ventilate a room by opening the windows.

So:

• das Zimmer lüften = to air out the room

This is a very common German verb. It does not mean simply to lift, even though it looks a bit similar to English words. It comes from Luft = air.

Because in German, a subordinate clause introduced by a conjunction like bevor is normally separated from the main clause by a comma.

So:

• Bevor wir das Regal montieren, müssen wir erst das Zimmer lüften.

That comma is required in standard German spelling.

Yes. You can also say:

• Wir müssen erst das Zimmer lüften, bevor wir das Regal montieren.

This means the same thing.

The difference is mainly one of focus:

• Starting with Bevor... puts emphasis on the time relationship
• Starting with Wir müssen... puts emphasis on the main action/obligation

Both are completely normal.

Because müssen is a modal verb. With modal verbs in German, the second verb usually appears in the infinitive at the end of the clause.

So:

• wir müssen ... lüften = we have to air out ...

The structure is:

• subject + modal verb + other elements + infinitive

Examples:

• Wir müssen gehen.
• Ich kann kommen.
• Sie will schlafen.

So in your sentence:

• müssen is the conjugated modal verb
• lüften stays in the infinitive at the end

In this sentence, the subject has to be expressed in both clauses.

• Bevor wir das Regal montieren
• müssen wir erst das Zimmer lüften

German does not normally omit the subject the way some languages do. Even though both clauses have the same subject, wir still needs to appear in each clause.

Yes. In German, it is a complete subordinate clause:

• Bevor = subordinating conjunction
• wir = subject
• das Regal = object
• montieren = verb

It depends on the main clause for the full sentence, but grammatically it is still a proper clause. English works similarly with Before we assemble the shelf... — it also feels like something is still coming, but it is still a clause.