Learn Hardcore French: Le courrier arrive à la maison.

du courrier would mean “some mail” (a partitive sense) and implies an unspecified quantity rather than the daily routine.
Dropping the article entirely (courrier arrive…) is ungrammatical in standard French, because most nouns need an article (definite, partitive, or indefinite) in a simple statement.

Is courrier masculine or feminine? How can I tell which article to use?

Courrier is a masculine noun, so it takes le in the singular. A quick tip: many French nouns ending in -ier are masculine (e.g., atelier, sommelier), but it’s always safest to check a dictionary when you’re unsure.

Is courrier countable or uncountable? Could I say un courrier?

In everyday French, courrier (mail) is treated as an uncountable/mass noun—you don’t normally use un. If you want to refer to one item, you’d say:

une lettre (a letter)
un colis (a parcel)
In formal or administrative language, you might see un courrier meaning “one piece of mail,” but in spoken French you’ll almost always use courrier without an article for quantity.

Why is the verb in the simple present arrive instead of “is arriving”? Does French have a continuous tense?

French does not have a distinct present‐continuous tense like English. The simple present (il arrive) covers both “he arrives” and “he is arriving.” If you want to stress that the action is in progress right now, you can use être en train de + infinitive (e.g., il est en train d’arriver), but for routine events the simple present is standard.

Why is it arrive à la maison? Could it be arrive dans la maison or another preposition?

To express arriving at a general destination, French uses arriver + à + article + place:

arriver à la maison = “arrive at home/at the house.”
If you said arriver dans la maison, you’d be specifying arriving inside the building (as if you were outdoors and then entered). The choice of à vs. dans changes the focus:
à → reaching a destination
dans → entering into an interior

Why is it à la maison and not au maison?

In French, au is the contraction of à + le. Since maison is feminine, you use à la (there is no contraction with la). If the noun were masculine (e.g., le bureau), you would say au bureau.

Could we use chez moi instead of à la maison?

Yes. Le courrier arrive chez moi literally means “the mail arrives at my place.”

chez
- person/pronoun denotes someone’s home or company (e.g., chez Paul, chez eux).
à la maison is more neutral/general, simply “at the house” or “at home.”

What’s the difference between arriver and venir? Can I say Le courrier vient à la maison?

Arriver means “to arrive” (to reach a destination).
Venir means “to come” (movement toward the speaker’s perspective).
You wouldn’t normally say Le courrier vient à la maison. Instead you might say Le facteur vient à la maison (“the postman comes to the house”), but for the mail itself you use arriver: Le courrier arrive à la maison.