Vor dem Einzug müssen wir noch streichen, und nach dem Auszug wollen wir alles sauber übergeben.

Because they are nouns here, not verbs.

• einziehen = to move in
• der Einzug = the move-in / moving in
• ausziehen = to move out
• der Auszug = the move-out / moving out

German often turns verbs into nouns to talk about an event or process. All nouns are capitalized, so you get Einzug and Auszug.

Because dem is the dative article for a masculine singular noun, and both Einzug and Auszug are masculine: der Einzug, der Auszug.

In this sentence:

• vor is used in a time sense = before
• nach is also a time preposition = after

With these meanings, they take the dative:

• vor dem Einzug
• nach dem Auszug

So this is not about motion here; it is just the normal case required by the prepositions.

Because German main clauses follow the verb-second rule.

The first position is taken by Vor dem Einzug, so the finite verb must come next:

• Vor dem Einzug | müssen | wir ...

If the sentence started with the subject, then you would get:

• Wir müssen vor dem Einzug noch streichen.

Both are correct. The version in your sentence simply puts emphasis on the time expression first.

Because they are paired with modal verbs:

• müssen
  • streichen
• wollen
  • übergeben

In a German main clause with a modal verb, the modal is the conjugated verb in second position, and the main verb stays in the infinitive at the end:

• wir müssen ... streichen
• wir wollen ... übergeben

That is a very common German sentence pattern.

Here noch means something like still, yet, or before then.

So:

• wir müssen noch streichen
  = we still have to paint
  = we haven’t painted yet, and it still needs to be done

It adds the idea that the action is not done yet.

Yes, in this context it does.

streichen has several meanings in German, including:

• to paint
• to stroke
• to cross out / cancel

But in the context of a home, apartment, or moving, streichen usually means to paint walls or repaint.

So here it means something like paint the place before moving in.

German often leaves the object out when it is obvious from context.

In a moving or apartment context, streichen usually implies the walls, the apartment, or the rooms. So speakers do not always say the object explicitly.

You could say:

• die Wohnung streichen
• die Wände streichen

But just noch streichen sounds very natural when everyone already knows what is being painted.

sauber übergeben means to hand over or leave something clean.

Here sauber is not placed directly before a noun, so it is not acting like an attributive adjective. Instead, it describes the condition or manner of the handover.

That is why it has no adjective ending.

Compare:

• die saubere Wohnung = the clean apartment

  • adjective before a noun, so it gets an ending

• die Wohnung sauber übergeben = hand over the apartment clean

  • sauber describes the result/condition, so no ending

Here alles means something like everything or the whole place / everything involved.

In context, it usually refers to the apartment, rooms, fixtures, and generally leaving everything in proper condition. It is a broad, practical way of saying that everything should be handed over clean.

A speaker could also be more specific and say:

• die Wohnung sauber übergeben

But alles sauber übergeben sounds natural if the speaker means the whole situation or all parts of the place.

No. In the sense of to hand over, übergeben is an inseparable verb.

That matters most in forms like the past participle:

• übergeben = handed over
• not übergegeben

In the present infinitive, you do not see a visible difference, so learners often wonder about it. But here it is the normal inseparable verb meaning to hand over.

Because the sentence links two main clauses:

• Vor dem Einzug müssen wir noch streichen
• nach dem Auszug wollen wir alles sauber übergeben

In German, a comma before und in this situation is often left out, but it can be used to make the structure clearer, especially when the clauses are a bit longer.

So both of these are acceptable:

• Vor dem Einzug müssen wir noch streichen und nach dem Auszug wollen wir alles sauber übergeben.
• Vor dem Einzug müssen wir noch streichen, und nach dem Auszug wollen wir alles sauber übergeben.

The comma here helps readability.