Bevor wir umziehen, räumen wir das Wohnzimmer aus.

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

So:

• Main clause: Wir ziehen um.
• Subordinate clause: bevor wir umziehen

That is a very common pattern with words like bevor, weil, dass, wenn, and obwohl.

In a German main clause, the finite verb normally has to be in second position.

Here, the whole clause Bevor wir umziehen takes the first position in the sentence. That means the verb of the main clause must come immediately after it:

Bevor wir umziehen, räumen wir das Wohnzimmer aus.

So the order is:

1. Bevor wir umziehen
2. räumen
3. wir
4. das Wohnzimmer
5. aus

If the subordinate clause came second, you would get the more familiar order:

Wir räumen das Wohnzimmer aus, bevor wir umziehen.

Because German normally uses a comma to separate a subordinate clause from a main clause.

Since bevor wir umziehen is a subordinate clause, it must be separated by a comma:

Bevor wir umziehen, räumen wir das Wohnzimmer aus.

This comma is required in standard German.

Both ausräumen and umziehen are separable verbs.

In a main clause, a separable verb splits:

• Wir räumen das Wohnzimmer aus.
• Wir ziehen um.

But in a subordinate clause, the verb parts stay together at the end:

• ..., bevor wir umziehen
• ..., weil wir das Wohnzimmer ausräumen

So umziehen is not really behaving differently here. It is just in a subordinate clause, where separable verbs stay together.

No, they are related but not the same.

• ausräumen = to clear out, to empty out
• aufräumen = to tidy up, to clean up

In this sentence, ausräumen suggests removing things from the living room, probably because you are moving.

So:

• Wir räumen das Wohnzimmer aus. = We clear out the living room.
• Wir räumen das Wohnzimmer auf. = We tidy up the living room.

That is an important difference.

Because Wohnzimmer is a neuter noun in German, so its basic article is das:

• das Wohnzimmer = the living room

Also, here it is the direct object of ausräumen, so it is in the accusative case. But for a neuter noun, the article is still das in both nominative and accusative singular.

So:

• Nominative: das Wohnzimmer
• Accusative: das Wohnzimmer

That is why the form does not change.

Because ausräumen takes the room itself as the thing being cleared out.

So:

• das Wohnzimmer ausräumen = to clear out the living room

If you say aus dem Wohnzimmer, that means out of the living room, which focuses on the place something is being removed from.

For example:

• Wir räumen die Möbel aus dem Wohnzimmer.
  = We remove the furniture from the living room.

But in your sentence, the idea is that the whole room is being emptied, so das Wohnzimmer is the direct object.

German very often uses the present tense to talk about the future when the context already makes the time clear.

Here, bevor and the situation of moving make the future meaning obvious:

Bevor wir umziehen, räumen wir das Wohnzimmer aus.

A natural English translation might use will or a present form depending on context, but German does not need werden here.

Using the present tense like this is extremely common in everyday German.

Yes, absolutely. That is also correct.

Both sentences mean basically the same thing:

• Bevor wir umziehen, räumen wir das Wohnzimmer aus.
• Wir räumen das Wohnzimmer aus, bevor wir umziehen.

The difference is mostly one of focus or style:

• Starting with Bevor wir umziehen puts the time frame first.
• Starting with Wir räumen das Wohnzimmer aus puts the action first.

Both are natural German.

Here it means to move house.

The verb umziehen can mean both:

• to move house
• to get changed

Context tells you which meaning is intended.

For example:

• Wir ziehen nächste Woche um. = We are moving next week.
• Ich ziehe mich um. = I am getting changed.

In your sentence, because of before we move and clearing out the living room, the meaning is clearly to move house.