Learn Hardcore German: Die Kinder sitzen an der Mauer und lesen ein Buch. - The children sit by the wall and read a book.

an der Mauer = at/by the wall (often next to it or leaning against it), focusing on proximity to a vertical surface.
auf der Mauer = on the wall, i.e., on top of it. Choose an if they are beside the wall, auf if they’re perched on top.

Why der in an der Mauer when the noun is die Mauer?

Because of case. an is a two-way preposition:

With location (no movement), it takes the dative case. Feminine dative singular of die Mauer is der. So: an der Mauer = at the wall (dative).

Could it be am Mauer?

No. am = an dem (only for masculine/neuter dative). Mauer is feminine, so you must use an der, not am.

When would it be an die Mauer instead?

Use accusative after an for movement towards a place:

sich an die Mauer setzen = to sit down at the wall (movement).
an der Mauer sitzen = to be sitting at the wall (location, dative).

What’s the difference between sitzen and (sich) setzen?

sitzen describes the state: to be sitting.
sich setzen describes the action: to sit down. Compare:
Die Kinder sitzen an der Mauer. (state)
Die Kinder setzen sich an die Mauer. (they move there and sit down)

German doesn’t have a progressive form—so is sitzen here like “are sitting”?

Yes. German uses the simple present for ongoing actions:

Die Kinder sitzen … und lesen … ≈ “The children are sitting … and are reading …” You can add gerade for emphasis on “right now,” or colloquially use the regional am-Progressiv: Die Kinder sind am Lesen (informal).

Why is it lesen and not liest?

Subject–verb agreement. Die Kinder is third-person plural, so:

sie lesen (they read)
Singular would be das Kind liest (the child reads).

Does lesen ein Buch mean they are reading one book together, or each has a book?

It’s ambiguous out of context. It can mean:

One book together (common reading), or
Each child has a book (distributive). To disambiguate:
Together: Die Kinder lesen gemeinsam ein Buch.
Each: Die Kinder lesen jeweils ein Buch / Jedes Kind liest ein Buch. If you clearly mean multiple books, say Die Kinder lesen Bücher.

Why is it ein Buch (not einen Buch)?

Buch is neuter (das Buch). Neuter accusative singular keeps ein unchanged:

Masculine accusative: einen (e.g., einen Film)
Neuter accusative: ein (e.g., ein Buch)
Feminine accusative: eine (e.g., eine Zeitung)

Why does the second verb come right after und: … und lesen ein Buch?

You have two coordinated main clauses sharing the same subject (Die Kinder). In the second clause, the subject is omitted but understood. The finite verb still appears early:

Die Kinder sitzen … und lesen ein Buch. This is the most natural order. You could also front the object for emphasis: … und ein Buch lesen (puts focus on the book).

Should there be a comma before und?

Not here. With a shared subject and two predicates, no comma is used:

Die Kinder sitzen … und lesen … If you repeat the subject—… , und sie lesen …—a comma becomes optional by modern rules.

Does an der Mauer apply to both actions (sitting and reading)?

By default, readers usually understand that both happen there. If you want to make it explicit, add a place word in the second part:

Die Kinder sitzen an der Mauer und lesen dort ein Buch.

Could I use bei or neben instead of an?

an: at/by, often touching or very close to a vertical surface.
neben: next to, side-by-side, without implying contact.
bei: near/at the vicinity of, looser proximity. Choose based on nuance: an der Mauer (by/against it), neben der Mauer (beside it), bei der Mauer (somewhere near it).

What’s the difference between Mauer and Wand?

Mauer: an exterior/outdoor wall, free-standing or boundary (e.g., die Berliner Mauer).
Wand: an interior wall inside a building. So outdoors you’d usually say Mauer, indoors Wand.

Why are Kinder, Mauer, and Buch capitalized?

All nouns in German are capitalized. Kinder, Mauer, Buch are nouns; und, an, der, ein are not.

How do I pronounce tricky parts like ie, au, and ch here?

ie in Kinder is a short i actually spelled i; the long sound is in words like vier. In Kinder, the vowel is short [ɪ].
au in Mauer is [aʊ] (like English “ow”).
ch in Buch is the back (like Scottish “loch”).

Can I swap the order: Die Kinder lesen ein Buch und sitzen an der Mauer?

Yes, it’s grammatical. It slightly shifts the flow/focus to the reading first, then adds where they are. The original highlights the location first, then the activity.

What are the past forms I’d need?

sitzen: Präteritum saßen; Perfekt haben gesessen.
lesen: Präteritum lasen; Perfekt haben gelesen. Example: Gestern saßen die Kinder an der Mauer und haben ein Buch gelesen.