Learn Hardcore German: Unser Musikgeschmack ist unterschiedlich, aber wir hören trotzdem zusammen Musik im Wohnzimmer. - Our taste in music is different, but we still listen to music together in the living room.

Musikgeschmack is masculine, singular, nominative (it’s the subject).
For masculine singular nominative, unser takes no extra ending: unser Musikgeschmack.

Compare:

unser Musikgeschmack – our taste in music (masc. nom. sg.)
unsere Musikgeschmäcker – our tastes in music (masc. nom. pl.)
unsere Wohnung – our apartment (fem. nom. sg.)
unser Auto – our car (neut. nom. sg.)

So unser Musikgeschmack is correct because there is one shared taste in music, grammatically masculine singular.

Why is it ist unterschiedlich and not something like ist unterschiedlicher or ist unterschiedlichen?

Here unterschiedlich is a predicate adjective after the verb sein (ist).

In German:

Adjectives before a noun take an ending:
- ein unterschiedlicher Musikgeschmack – a different taste in music
Adjectives after verbs like sein, werden, bleiben usually do not take endings:
- Unser Musikgeschmack ist unterschiedlich. – Our taste in music is different.

So:

ist unterschiedlich = “is different” (correct)
Forms like ist unterschiedlicher / ist unterschiedlichen are wrong in this structure.

Could I say verschieden instead of unterschiedlich? Is there a difference?

In this sentence, unterschiedlich and verschieden are basically synonyms. Both mean “different”.

Unser Musikgeschmack ist unterschiedlich.
Unser Musikgeschmack ist verschieden.

Both are fine and common. Subtle tendencies (not rules):

unterschiedlich sometimes highlights variation / differences between things.
verschieden is also “different” or “various”, sometimes used like “several”:
- Ich habe verschiedene Bücher. – I have several/different books.

But for “our taste in music is different”, both are natural.

What is the function of aber here, and how is it different from sondern?

Aber is a coordinating conjunction meaning “but”. It introduces a contrast:

Unser Musikgeschmack ist unterschiedlich, aber …
= Our taste in music is different, but …

Sondern also translates as “but rather”, but it’s used only after a negation to correct or replace something:

Nicht Tee, sondern Kaffee. – Not tea, but (rather) coffee.
Er ist nicht Arzt, sondern Lehrer. – He’s not a doctor, but a teacher.

In your sentence, there is no negation in the first part, so sondern would be wrong. You must use aber.

What does trotzdem add here, and how is it different from obwohl?

Trotzdem is an adverb meaning roughly “nevertheless”, “nonetheless”, “anyway”, “despite that”.

Unser Musikgeschmack ist unterschiedlich, aber wir hören trotzdem zusammen Musik …
= Our taste in music is different, but we still / nevertheless listen to music together …

Obwohl is a subordinating conjunction meaning “although” / “even though”, and it introduces a subordinate clause:

Obwohl unser Musikgeschmack unterschiedlich ist, hören wir zusammen Musik.
= Although our taste in music is different, we listen to music together.

So:

trotzdem = adverb, stands in the main clause.
obwohl = conjunction, starts a subordinate clause and sends the verb to the end.

Your sentence uses aber + trotzdem, which is very natural: “but nevertheless”. You could also say:

Unser Musikgeschmack ist unterschiedlich. Trotzdem hören wir zusammen Musik.
Trotzdem hören wir zusammen Musik im Wohnzimmer.

All are idiomatic, just slightly different in emphasis and rhythm.

Why is the word order wir hören trotzdem zusammen Musik, and could trotzdem come first?

In the given sentence:

wir = subject (position 1)
hören = conjugated verb (position 2)
trotzdem = adverb inside the middle field

German main clauses require the finite verb in second position, but adverbs like trotzdem can move around a bit as long as the verb stays second.

Possible word orders (all correct, with slightly different emphasis):

Wir hören trotzdem zusammen Musik im Wohnzimmer.
Wir hören zusammen trotzdem Musik im Wohnzimmer.
Trotzdem hören wir zusammen Musik im Wohnzimmer. (here trotzdem is in the “Vorfeld” as position 1)

The important rules:

The conjugated verb = always in second position in a main clause.
Elements like trotzdem, zusammen, im Wohnzimmer can be reordered for emphasis or style.

Why is there no article before Musik in wir hören … Musik?

In German, Musik is often treated like an uncountable or abstract noun when you mean music in general. In that case, you normally don’t use an article:

Wir hören Musik. – We listen to music.
Ich mache gern Musik. – I like making music.

If you make it specific, you can add an article or another determiner:

Wir hören die Musik. – We (are) listen(ing) to the music. (some specific music, e.g. from the next room)
Wir hören seine Musik. – We listen to his music.
Wir hören diese Musik. – We listen to this music.

In your sentence, it means music in general that they listen to together, so no article is natural.

Why is zusammen placed before Musik? Could I move it?

Zusammen means “together” and is an adverb modifying hören (to listen). It answers “How do we listen?” → together.

In the sentence:

wir hören trotzdem zusammen Musik …

word order is:

Subject: wir
Verb: hören
Adverbs: trotzdem zusammen
Object: Musik

You can move zusammen somewhat, but it usually stays near the verb it modifies:

Wir hören zusammen Musik. (very natural)
Wir hören Musik zusammen. (also possible, just a bit different rhythm)

Putting zusammen right before Musik is common and sounds natural. It still clearly refers to hören, not to the noun Musik.

What does im in im Wohnzimmer stand for?

Im is the contraction of in dem:

in = in
dem = the (dative singular, masculine/neuter)
in dem Wohnzimmer → im Wohnzimmer

So im Wohnzimmer literally means “in the living room”.

These contractions are very common in spoken and written German:

in dem → im
an dem → am
bei dem → beim
zu dem → zum
zu der → zur

Why is it im Wohnzimmer (dative) and not accusative? Which case does in take here?

The preposition in can take dative or accusative, depending on whether it expresses:

Location (where?) → dative
Direction / movement (where to?) → accusative

Your sentence describes where they are listening to music (a location), not movement:

Wir hören … Musik im Wohnzimmer. – We listen to music in the living room.
→ where? → location → dative

So:

im Wohnzimmer = in dem Wohnzimmer (dative)

Compare:

Wir gehen ins Wohnzimmer. – We go into the living room. (in das, accusative, movement)
Wir sind im Wohnzimmer. – We are in the living room. (in dem, dative, location)