Learn Hardcore German: In der Musikschule lernt meine Tochter Flöte, und ihr Bruder spielt Trommel im Schulorchester. - At the music school my daughter learns the flute, and her brother plays the drum in the school orchestra.

Im is a contraction of in dem, which is dative masculine/neuter, so it cannot be used with the feminine noun Musikschule.
In die Musikschule would mean “into the music school / to the music school” (movement toward it), which is not the meaning here.

Why is the word order In der Musikschule lernt meine Tochter Flöte and not In der Musikschule meine Tochter lernt Flöte?

German main clauses follow the verb-second (V2) rule: the finite verb (here: lernt) must be the second element in the sentence.

If you start with a prepositional phrase like In der Musikschule, that occupies the first position. The verb must then come next, and the subject follows after the verb:

In der Musikschule (1st position)
lernt (2nd position: finite verb)
meine Tochter Flöte (rest of the sentence)

So:

Correct: In der Musikschule lernt meine Tochter Flöte.
Also correct: Meine Tochter lernt in der Musikschule Flöte. (Here meine Tochter is first, lernt still second.)
Incorrect: In der Musikschule meine Tochter lernt Flöte. (The verb is in third position.)

Why is there no article before Flöte? Why not meine Tochter lernt die Flöte?

With musical instruments, German very often omits the article when they are objects of spielen or lernen:

Flöte spielen / Flöte lernen
Klavier spielen / Klavier lernen
Gitarre spielen / Gitarre lernen

Using the article here (die Flöte spielen, die Gitarre spielen) is possible but tends to sound more concrete, like referring to a specific physical instrument in a context, not the skill in general.

So:

meine Tochter lernt Flöte ≈ my daughter is learning (to play) the flute (the skill)
meine Tochter lernt die Flöte can sound like she is learning this particular flute (less usual for the general meaning).

Why can you say lernt … Flöte without spielen, instead of lernt Flöte zu spielen?

In German, with instruments, lernen + Instrument is a standard shorthand for “lernen, [Instrument] zu spielen”:

Sie lernt Flöte. = Sie lernt, Flöte zu spielen.
Er lernt Klavier. = Er lernt, Klavier zu spielen.

Both forms are grammatically correct. The version without spielen is shorter and more common in everyday language when talking about learning instruments. The version with zu spielen sounds a bit more explicit or formal, but not wrong.

Why is it meine Tochter but ihr Bruder (not ihre Bruder)?

Both meine and ihr are possessive determiners, and their form depends on the gender and case of the noun they modify, not directly on who owns the thing.

Tochter is feminine, nominative:
- Feminine nominative singular → meine Tochter
Bruder is masculine, nominative:
- Masculine nominative singular → ihr Bruder

If Bruder were accusative, you would get ihren Bruder; if it were feminine, you'd see ihre; but here Bruder is the subject of the second clause, so it is nominative masculine and takes ihr.

What exactly does ihr Bruder mean here, and who does ihr refer to?

In this sentence, ihr Bruder means “her brother” and ihr refers back to meine Tochter:

meine Tochter (she) → ihr Bruder (her brother)

Grammatically, ihr as a possessive can mean:

her (3rd person singular feminine)
their (3rd person plural)
your (formal, singular or plural)

So, outside of context, ihr Bruder could mean “her brother,” “their brother,” or “your brother (formal).” Here, because the only obvious person mentioned before is meine Tochter, the natural reading is “her brother”, i.e. the daughter’s brother.

Is spielt Trommel idiomatic German, or should it be spielt die Trommel or spielt Schlagzeug?

All three can appear, but they’re not equally common:

For the general activity of playing drums as an instrument, the most idiomatic is:
- Er spielt Schlagzeug. = He plays the drums / percussion.
spielt die Trommel focuses more on a specific drum (one physical instrument).
spielt Trommel (without article) is understandable and may appear in some contexts, but for standard modern German, people would normally say Schlagzeug spielen for “playing drums” as a musical activity.

So a very natural version of the sentence would be:
… und ihr Bruder spielt Schlagzeug im Schulorchester.

Why is there a comma before und in …, und ihr Bruder spielt Trommel im Schulorchester.?

In German, when two independent main clauses are joined by und, a comma is required:

In der Musikschule lernt meine Tochter Flöte,
und ihr Bruder spielt Trommel im Schulorchester.

Each part has its own subject and finite verb:

1st clause: meine Tochter – lernt
2nd clause: ihr Bruder – spielt

Because both are full main clauses, German orthography rules say you must use a comma before und here. (In English, the comma would be optional or stylistic; in German it’s obligatory in this sentence type.)

What does im Schulorchester mean exactly, and why im instead of in dem or in der?

Im is the standard contraction of in dem:

in + dem → im

You use dem because Orchester is neuter:

das Orchester (nominative)
dem Orchester (dative)

The whole noun is a compound: das Schulorchester (the school orchestra). With in expressing location (“in the school orchestra”), you again need the dative:

in dem Schulorchester → contracted to im Schulorchester

In der would be dative feminine, but Orchester is neuter, so in der Schulorchester would be incorrect.

Why are Musikschule and Schulorchester written as single words and not as Musik Schule or Schul Orchester?

German regularly forms compound nouns by writing several nouns together as one word:

Musik
- Schule → Musikschule (music school)
Schule
- Orchester → Schulorchester (school orchestra)

Writing them separately (like Musik Schule) would either be wrong or change the meaning/feeling. In German spelling, the default is: if two nouns belong together as one concept, you write them together as a compound.

Which cases are used in this sentence, and how do they match the endings der, meine, ihr, Trommel, etc.?

Main cases in the sentence:

In der Musikschule
- Musikschule is feminine dative (location with in)
- Article: der (feminine dative singular)
meine Tochter
- Subject of the first clause → nominative
- Feminine nominative singular → meine Tochter
Flöte
- Direct object of lernt → accusative
- Feminine accusative singular often looks like nominative (no special ending): Flöte
ihr Bruder
- Subject of the second clause → nominative
- Masculine nominative singular with possessive → ihr Bruder
Trommel
- Direct object of spielt → accusative
- Feminine accusative singular again same form: Trommel
im Schulorchester
- Location with in → dative
- Neuter dative singular: dem Orchester → im Orchester → im Schulorchester

So you see nominative for the subjects (meine Tochter, ihr Bruder), accusative for the instruments being learned/played (Flöte, Trommel), and dative after in for places (der Musikschule, im Schulorchester).