Learn Hardcore Italian: Quando entro in casa, starnutisco per la polvere. - When I come into the house, I sneeze because of the dust.

With the verb entrare (to enter), Italian uses the preposition in for enclosed places: entrare in casa, entrare in ufficio, entrare in chiesa.
Use a casa with verbs of going/being at home: vado a casa, torno a casa, sono a casa.
So: entro in casa is correct; entro a casa is not.

What’s the difference between in casa and a casa?

in casa = inside the house/indoors.
a casa = at home/to home (location or destination, not the idea of crossing the threshold).
Your sentence describes going inside, so in casa fits.

Do I need an article with casa here? Why not nella casa?

No article is needed in the set phrase in casa (it’s idiomatic and general).
Use nella casa only when you mean a specific house that’s been identified: Entro nella casa gialla = I enter the yellow house. For your own home in general, in casa is normal.

Is the comma after the Quando clause necessary?

It’s standard and recommended when the dependent clause comes first: Quando..., [comma] .... If you swap the order, a comma is usually not needed: Starnutisco quando entro in casa.

Why is the present tense used? How would I talk about a single future occurrence?

Here the present expresses a habitual action. For a single future event, Italian typically uses the future in both clauses:

Quando entrerò in casa, starnutirò.
In everyday speech, you may also hear the future only in the main clause or even the present in the time clause, but the safest neutral form is future in both.

Can I use mentre, appena, or ogni volta che instead of quando?

mentre = while (overlapping actions): not ideal here.
appena = as soon as: emphasizes immediacy: Appena entro in casa, starnutisco.
ogni volta che = every time that: stresses repeated occurrence: Ogni volta che entro in casa, starnutisco.

What exactly is starnutisco? How do I conjugate starnutire?

Starnutisco is 1st person singular present of starnutire (to sneeze), an -ire verb of the -isco type.
Present indicative:

io starnutisco, tu starnutisci, lui/lei starnutisce
noi starnutiamo, voi starnutite, loro starnutiscono

Is starnuto a verb? Can I say it instead?

Starnuto is a noun (a sneeze), not a verb.

Verb: starnutire → Starnutisco spesso.
Noun: fare uno starnuto → Ho fatto uno starnuto.

Why per la polvere? Could I say a causa della polvere or per via della polvere?

All three express cause:

per la polvere = neutral, very common in speech.
a causa della polvere = a bit more formal/explicit.
per via della polvere = informal/colloquial feel.
They’re all fine here.

Why is there an article in la polvere? Could I omit it?

Italian usually uses the definite article with mass nouns when specifying the cause: per la polvere, per il freddo.
Omitting it (per polvere) sounds unnatural in this context.

Can I use perché instead of per to express the cause?

Use perché to introduce a clause (a full sentence):

Starnutisco perché c’è polvere.
With a bare noun, use per/a causa di:
Starnutisco per la polvere.
So not: ✗ Starnutisco perché la polvere.

Could I say dalla polvere instead of per la polvere?

Not for cause. Da with nouns tends to mean from/against (origin or protection): mi proteggo dalla polvere.
For cause, use per or a causa di: starnutisco per la polvere.

Can I change the word order to start with the main clause?

Yes: Starnutisco per la polvere quando entro in casa.
Meaning stays the same. Punctuation also becomes simpler (no comma needed).

Could I use rientrare instead of entrare?

rientrare = go back in/return.
If you mean you’re coming back home, you can say: Rientro a casa (I return home) or, if you emphasize going back inside, Rientro in casa.
If it’s just crossing the threshold (not necessarily returning), Entro in casa is the neutral choice.

Is there another natural way to express “I feel like sneezing” here?

Yes: Mi viene da starnutire per la polvere.
This focuses on the urge rather than the completed action.