Vor dem Auszug putzen wir die Küche gründlich.

Because vor is a two-way preposition, and here it expresses time: before the move-out.
When vor is used in a time expression, it takes the dative case.

• der Auszug = the move-out / moving out
• dative singular of der = dem

So:

• der Auszug → nominative
• vor dem Auszug → dative after vor

Here, der Auszug means moving out of a place, especially a home or apartment.

It is a noun formed from ausziehen, which can mean to move out in the housing sense.

A useful distinction:

• der Auszug = moving out / departure from a residence
• der Umzug = the move itself, especially changing residence from one place to another

So in this sentence, Vor dem Auszug means before moving out or before the move-out.

Because all nouns are capitalized in German.

So:

• ausziehen = verb
• der Auszug = noun

That is why you see Auszug with a capital A.

Yes, this is completely normal German word order.

German often puts a time expression at the beginning of the sentence for emphasis or as the topic:

• Vor dem Auszug putzen wir die Küche gründlich.

When something other than the subject comes first, the finite verb still stays in second position. This is a very important German rule.

So the structure is:

• Vor dem Auszug = element 1
• putzen = finite verb in position 2
• wir = subject after the verb

You could also say:

• Wir putzen die Küche vor dem Auszug gründlich.

That is also grammatical, but the original sentence puts more focus on when it happens.

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

Since Vor dem Auszug is placed first, the verb has to come next:

• Vor dem Auszug | putzen | wir ...

This is often called inversion from an English-speaking learner’s perspective, although in German it is just the normal word-order rule.

Compare:

• Wir putzen die Küche gründlich.
• Vor dem Auszug putzen wir die Küche gründlich.

Both are correct; the second one simply begins with a time phrase.

Because die Küche is the direct object of putzen.

The verb putzen takes an object: you clean something.

• die Küche is feminine
• nominative singular: die Küche
• accusative singular: die Küche

For feminine nouns, the article is the same in nominative and accusative, so you still see die.

Compare with a masculine noun, where the article would change:

• der Boden → nominative
• den Boden → accusative

Gründlich means thoroughly.

In this sentence it functions as an adverb, describing how the kitchen is cleaned.

German often places this kind of adverb later in the sentence, especially after the object:

• Wir putzen die Küche gründlich.

That sounds very natural.

You may also hear slightly different placements in other contexts, but here ... die Küche gründlich is a standard and idiomatic order.

Yes, putzen is very natural here.

For everyday cleaning, German commonly uses:

• putzen = to clean, especially in a practical household sense
• sauber machen = to make clean / clean up
• reinigen = to clean, but often sounds more formal, technical, or less everyday

So:

• Vor dem Auszug putzen wir die Küche gründlich.
  sounds natural and everyday.

Reinigen would be grammatically possible, but less typical in ordinary conversation about household chores.

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

Here, Vor dem Auszug already tells you when it will happen, so the present tense is enough:

• Vor dem Auszug putzen wir die Küche gründlich.

This is very common in German.

English often prefers will or going to, but German usually does not need a separate future form here.

Yes, absolutely. That is a very natural alternative.

The difference is mainly structural:

• Vor dem Auszug ... uses a noun phrase
• Bevor wir ausziehen, ... uses a subordinate clause

So:

• Vor dem Auszug putzen wir die Küche gründlich.
• Bevor wir ausziehen, putzen wir die Küche gründlich.

Both mean essentially the same thing.

A small nuance:

• Vor dem Auszug sounds a bit more compact and noun-based.
• Bevor wir ausziehen sounds a bit more verbal and explicit.

Yes. Auszug can have several meanings depending on context, for example:

• moving out
• an excerpt from a text
• a withdrawal or extraction in some technical contexts

But in this sentence, because of Vor dem Auszug and the context of cleaning the kitchen, the meaning is clearly moving out of a home or apartment.

This is very common in German: one noun can have several meanings, and context tells you which one is intended.