Learn Hardcore German: Ich wische den Staub weg, damit die Mikrowelle und das Spülbecken sauber bleiben. - I wipe away the dust so that the microwave and the sink stay clean.

Because Staub is the direct object of wischen (to wipe). Direct objects take the accusative case, and the masculine accusative article is den.

Nominative: der Staub (subject)
Accusative: den Staub (direct object)
Dative: dem Staub (indirect object; not used here)

Is Ich wische den Staub weg idiomatic? Could I say something else?

It’s correct and understandable. Common alternatives:

Very idiomatic for “I dust”: Ich wische Staub.
Also common: Ich wische den Staub ab.
Another verb: Ich staube ab. (to dust)
More general: Ich putze. (I’m cleaning) Adding weg emphasizes removal (“wipe away”), which fits well with dust.

What is weg doing at the end of the main clause?

weg- is a separable prefix. In main clauses, the finite verb stays in 2nd position and the prefix goes to the end:

Main clause: Ich wische den Staub weg.
In subordinate clauses and non-finite forms, the prefix attaches to the verb:
Subclause: …, weil ich den Staub wegwische.
Perfect: Ich habe den Staub weggewischt.
With zu: … den Staub wegzuwischen.

Why is there a comma before damit?

Because damit introduces a subordinate clause. In German, subordinate clauses are always set off by a comma.

What does damit mean here, and how is it different from um … zu?

damit introduces a purpose clause (“so that / in order that”) and takes a full finite verb:

Ich wische den Staub weg, damit die Mikrowelle und das Spülbecken sauber bleiben.

um … zu also expresses purpose but is only used when the subject of both clauses is the same. Since the purpose here concerns the sink and microwave (not “I”), you’d rephrase the goal:

Ich wische den Staub weg, um die Mikrowelle und das Spülbecken sauber zu halten.
Using um … zu with bleiben would wrongly imply “I stay clean.”

Why is the verb bleiben at the end of the damit-clause?

In subordinate clauses, German sends the finite verb to the end. Here the predicate adjective sauber comes before the verb, and bleiben is final:

…, damit [Subj] [Prädikativ] bleiben.

Why is it bleiben and not bleibt?

The subject of the subordinate clause is plural: die Mikrowelle und das Spülbecken (two items). So the verb is 3rd person plural: bleiben.

Singular: Die Mikrowelle bleibt sauber.
Plural: Die Mikrowelle und das Spülbecken bleiben sauber.

Why is it sauber and not saubere?

After linking verbs like sein, werden, and bleiben, adjectives are used predicatively and are not declined: ist/bleibt sauber.
They only take endings when attributive (before a noun):

ein sauberes Spülbecken, die saubere Mikrowelle vs. Das Spülbecken bleibt sauber.

Can I replace “die Mikrowelle und das Spülbecken” with a pronoun?

Yes: …, damit sie sauber bleiben.
Here sie = “they” (3rd person plural). The plural verb bleiben makes it clear it’s “they,” not “she.”

What are the genders and articles for these nouns?

die Mikrowelle (feminine; singular nominative: die; accusative: die)
Indefinite: eine Mikrowelle; plural: die Mikrowellen
das Spülbecken (neuter; singular nominative: das; accusative: das)
Indefinite: ein Spülbecken; plural: die Spülbecken
der Staub (masculine; accusative: den Staub)

Could I say sodass or dass instead of damit?

sodass/so dass expresses a result (“so that/as a result”):
Ich wische den Staub weg, sodass alles sauber bleibt. (This is what happens as a consequence.)
damit expresses intended purpose:
…, damit … sauber bleiben.
Plain dass isn’t used to express purpose; it’s a neutral complementizer.

Why use bleiben and not werden or halten?

bleiben = stay/remain clean (maintain a clean state)
werden = become clean (change into a clean state)
sauber halten = keep clean (emphasizes your maintaining action)
Choose based on meaning:
Prevent getting dirty: …, damit sie sauber bleiben.
Make them clean again: …, damit sie wieder sauber werden.
Focus on your effort: …, um sie sauber zu halten.

Is Ich wische den Staub (without weg) okay?

It’s understandable but less idiomatic than either

Ich wische Staub (very idiomatic), or
Ich wische den Staub ab/weg (explicitly “wipe off/away”).
Without ab/weg, wischen sounds unfinished unless you add what/where you’re wiping.

What’s the nuance between wegwischen, abwischen, and putzen?

wegwischen: wipe something away/remove it (often a substance: dust, spill, chalk).
abwischen: wipe something off a surface (the action is on the surface: table, sink, microwave).
putzen: clean (broad, any method).
With dust, you’ll hear both Staub abwischen and Staub wegwischen; for surfaces, etwas abwischen is especially common.

Why is die Mikrowelle singular if die can mean plural?

Here die is the feminine singular article. You can tell it’s singular because Mikrowelle is singular and it’s joined with another singular noun by und to form a plural subject. Plurals would also use die, but the noun would be plural: die Mikrowellen.

Do I need to capitalize these words?

Yes, all nouns are capitalized in German: der Staub, die Mikrowelle, das Spülbecken. Adjectives like sauber are not capitalized unless at the start of a sentence.

Any quick pronunciation tips for tricky parts?

w sounds like English “v”: wische, weg, Mikrowelle.
ich has the soft “ch”
ü in Spülbecken is rounded front vowel [y], not like English “u.”
Stress: MÍ-kro-wel-le, SPÜL-bec-ken, STAUB, wÍ-sche, da-MÍT.