Vor dem Einzug möchten wir die Wände streichen, damit beim Auszug alles ordentlich aussieht.

Because vor takes the dative here.

• vor + dative is used when it means before in a time sense.
• der Einzug becomes dem Einzug in the dative singular.

So:

• vor dem Einzug = before moving in / before the move-in

If vor describes a physical location, it can take either dative or accusative depending on whether there is movement, but in this sentence it is about time, so it must be dative.

Der Einzug literally means the moving in or move-in.

It comes from the verb einziehen, which can mean:

• to move into a place
• sometimes to collect or to withdraw, in other contexts

Here, Einzug is a noun meaning the event of moving into a home or apartment.

So:

• vor dem Einzug = before moving in
• beim Auszug = when moving out / upon moving out

Notice the nice pair:

• Einzug = moving in
• Auszug = moving out

This is standard verb-second word order in a German main clause.

The sentence begins with Vor dem Einzug, which is the first element. In a normal German main clause, the finite verb must come next, in the second position.

So the structure is:

• Vor dem Einzug = position 1
• möchten = position 2
• wir = subject after the verb

That gives:

• Vor dem Einzug möchten wir ...

If you start the sentence with the subject instead, then you get:

• Wir möchten vor dem Einzug die Wände streichen.

Both are correct; the version in your sentence emphasizes the time.

Both can often be translated as want to, but they feel different.

• wir wollen = more direct, stronger, firmer
• wir möchten = more polite, softer, more like we would like to

So:

• Wir wollen die Wände streichen. = We want to paint the walls.
• Wir möchten die Wände streichen. = We would like to paint the walls.

In many everyday situations, möchten sounds more natural and less forceful.

Because Wände is the plural of die Wand (wall), and here it is the direct object of streichen.

• singular: die Wand
• plural: die Wände

In the plural, nominative and accusative often look the same:

• die Wände

Since streichen here means to paint, you paint something, so that thing is the direct object:

• die Wände streichen = to paint the walls

No. Streichen has several meanings depending on context.

Common meanings include:

• to paint
• to stroke
• to cross out
• to spread (as in butter on bread)

In this sentence, because the object is die Wände, it clearly means:

• to paint the walls

This is a very common use in home/apartment contexts.

Damit introduces a purpose clause. It means something like:

• so that
• in order that

So the sentence says:

• We want to paint the walls so that everything looks tidy when we move out.

German often uses damit when the sentence explains the purpose of an action.

Compare:

• Wir streichen die Wände, damit alles ordentlich aussieht. = We paint the walls so that everything looks tidy.

A similar structure is um ... zu, but damit is especially common when the clause has its own subject or when the speaker simply chooses that structure.

Because damit introduces a subordinate clause, and in German subordinate clauses the conjugated verb usually goes to the end.

Main clause:

• Vor dem Einzug möchten wir die Wände streichen

Subordinate clause:

• damit beim Auszug alles ordentlich aussieht

Inside that subordinate clause:

• beim Auszug = time phrase
• alles = subject
• ordentlich = adjective/adverb-like predicate word
• aussieht = finite verb at the end

This is one of the most important German word-order patterns to learn.

Beim is a contraction of:

• bei dem → beim

So:

• beim Auszug = bei dem Auszug

In this sentence, beim Auszug means something like:

• when moving out
• at the time of moving out

Bei often means at, with, or during/in the context of, depending on the sentence.

Here it is used in a time-related way.

Because the sentence is expressing two slightly different relationships:

• vor dem Einzug = before moving in
• beim Auszug = during/when moving out

So the speaker means:

1. Before moving in, we want to paint the walls,
2. so that when we move out, everything looks tidy.

German uses different prepositions because the meanings are different:

• vor = before
• bei / beim = during, at the time of

You could not simply swap them without changing the meaning.

Because alles is grammatically singular.

Even though alles refers to everything, German treats it as a singular pronoun. So the verb is singular too:

• alles aussieht = correct
• alle aussehen = incorrect here

Also, alle means everyone or all (people/things) in a different way, but that is not what is meant here.

Examples:

• Alles ist gut. = Everything is good.
• Alles sieht ordentlich aus. = Everything looks tidy.

The verb is aussehen, which is a separable verb.

In a normal main clause, separable verbs split:

• Alles sieht ordentlich aus.

Here:

• sieht = the conjugated part
• aus = the separable prefix

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

• ..., damit alles ordentlich aussieht.

So:

• main clause: sieht ... aus
• subordinate clause / infinitive: aussieht, aussehen

This is a very common pattern with separable verbs.

Here ordentlich means:

• tidy
• neat
• properly kept
• sometimes in good order

So alles ordentlich aussieht means:

• everything looks tidy
• everything appears neat and in order

In other contexts, ordentlich can also mean things like proper, decent, or even quite a lot in colloquial German, but here the meaning is clearly about the apartment looking neat.

Yes, a very natural alternative is:

• Vor dem Einzug möchten wir die Wände streichen, um beim Auszug alles ordentlich aussehen zu lassen.

But this version is a bit more complex and may sound slightly more formal or stylistically heavier.

The original with damit is straightforward and very natural:

• ..., damit beim Auszug alles ordentlich aussieht.

In general:

• damit = very common, clear, flexible
• um ... zu = often used when the subject is the same in both clauses

Since the original sentence is simple and idiomatic, damit is an excellent choice.

It can mean other things in other contexts.

Common meanings of Auszug include:

• moving out / move-out
• extract / excerpt from a text
• withdrawal in some specialized contexts

But in housing or apartment contexts, Auszug usually means:

• moving out

So in this sentence:

• beim Auszug definitely means when moving out

Context makes the meaning clear.