Learn Hardcore German: Ich sortiere den Stapel Briefe nach Datum und Absender. - I sort the pile of letters by date and sender.

Because den is the masculine accusative article. Stapel (stack/pile) is masculine (der Stapel), and here it’s the direct object of sortieren, so it must be accusative: den Stapel. Quick reminder (singular): der (nom), den (acc), dem (dat), des (gen).

What’s going on with den Stapel Briefe? Why isn’t it den Stapel von Briefen?

German often uses a “bare genitive” after measure/quantity nouns: ein Stapel Briefe, ein Glas Wasser, ein Haufen Bücher. All mean “a pile/glass/heap of …” and don’t need von. You can say ein Stapel von Briefen too; it’s fine and a bit more colloquial/explicit. Written German often prefers the bare form; spoken German varies.

Is Briefe genitive or nominative here? It looks nominative plural.

It’s the genitive plural in a bare construction after a quantity noun, but without an article the form looks the same as nominative plural: Briefe. With an article, you’d see the genitive: ein Stapel der Briefe (rare/stylistic). More natural is: ein Stapel Briefe, ein Stapel alter Briefe (note the genitive plural adjective: alter).

Could I just say den Briefstapel instead of den Stapel Briefe?

Yes. Der Briefstapel (compound noun) is very common and concise. All of these are acceptable, with small stylistic differences:

den Briefstapel (compound)
den Stapel Briefe (bare genitive)
den Stapel von Briefen (with von)

Does nach here mean “after”? I thought nach = after/to.

Nach has several meanings. Besides “after” (temporal) and “to/toward” (with cities/countries, e.g., nach Berlin), it also means “according to/by” when expressing a criterion: nach Datum, nach Größe, nach Farbe. Here it means “by/according to.”

Are Datum and Absender in the dative after nach? I can’t see any endings.

Yes, nach always takes dative. There’s no visible ending because there’s no article. If you add articles, the dative shows: nach dem Datum und dem Absender. In this generic “sorting by criterion” sense, German usually omits the articles.

Should I repeat nach before both nouns: nach Datum und nach Absender?

You don’t need to; the preposition scopes over both: nach Datum und Absender. Repeating it (nach Datum und nach Absender) is also correct and can add clarity or emphasis, but it’s not required.

Why is Ich sortiere used for “I am sorting”? Where’s the progressive?

German normally uses the simple present for ongoing actions: Ich sortiere … covers “I sort” and “I am sorting.” Colloquial alternatives exist (e.g., Ich bin am Sortieren, Ich bin dabei, … zu sortieren), but the standard, most neutral form is just the present.

Why does the prepositional phrase nach Datum und Absender go at the end? Could it go elsewhere?

Objects usually come before adverbials, and this PP expresses the sorting criterion/manner, which naturally falls later: Ich sortiere [Object] [PP]. You can move it for emphasis or style: Ich sortiere nach Datum und Absender den Stapel Briefe, but the given order is the most neutral.

Do Datum and Daten both mean “date(s)”? I’ve seen Daten meaning “data.”

Das Datum = “the date.” Its plural is die Daten, which often means “data,” but it can also mean “dates” (calendar dates) in context. In this sentence we use the generic criterion: nach Datum = “by date” (no plural needed). If you needed plural explicitly, you might say nach den Daten or rephrase (nach den Datumsangaben, nach den Terminen) to avoid ambiguity.

What’s the difference between Absender and Sender? And what’s the word for the person who receives?

Absender = the sender of mail/letters (the one who sends).
Sender can mean “broadcaster/transmitter” or a sender in a technical sense; for mail, Absender is the usual word.
The receiver is the Empfänger.

Do I need commas around und in nach Datum und Absender?

No. Simple two-item coordination with und takes no comma: nach Datum und Absender. Commas appear in longer lists or where grammar requires them, but not here.

Can sortieren take other prepositions? What’s the difference between nach and in?

sortieren nach + Dativ = “sort by (criterion),” e.g., nach Datum, nach Größe.
sortieren in + Akkusativ = “sort into (categories/containers),” e.g., in Ordner, in Kisten. Related verbs: einsortieren (file/put away), aussortieren (weed out).

Why is there no article before Briefe?

Because Briefe here is the complement to the quantity noun Stapel in a bare genitive construction. You normally don’t use an article there: ein Stapel Briefe, zwei Stapel Rechnungen, ein Haufen Bücher.

What gender and plural does Stapel have?

Stapel is masculine: der Stapel. Its plural is identical: die Stapel. In the accusative singular, it’s den Stapel.