Sie zeigt Interesse an Musik.

In this sentence, Sie most naturally means she.

German sie/Sie can mean several things depending on context:

• sie = she
• sie = they
• Sie = you (formal)

Because the rest of the sentence is zeigt (singular form), Sie here is understood as she or formal you, not they. Since you said the meaning is already given, this sentence is probably intended as She shows interest in music.

A key detail:

• At the beginning of a sentence, both sie and Sie are written Sie, so capitalization alone does not help here.

Because the verb has to agree with the subject.

The infinitive is zeigen = to show.

With sie meaning she, the conjugation is:

• ich zeige
• du zeigst
• er/sie/es zeigt
• wir zeigen
• ihr zeigt
• sie/Sie zeigen

So:

• Sie zeigt Interesse an Musik. = She shows interest in music.

If the subject were they or formal you, it would be:

• Sie zeigen Interesse an Musik.

That is just the fixed German pattern.

In German, the noun Interesse is commonly used with an:

• Interesse an Musik
• Interesse an Politik
• Interesse an Kunst

So although English says interest in music, German says literally interest at/in relation to music with an.

This is something you usually just have to learn as a collocation:

• Interesse an + dative

A closely related verb is:

• sich interessieren für + accusative
  = to be interested in

For example:

• Sie zeigt Interesse an Musik.
• Sie interessiert sich für Musik.

Both are natural, but they use different prepositions.

Here an takes the dative.

The expression is:

• Interesse an + dative

So you would get forms like:

• Interesse an Musik
• Interesse an der Musik
• Interesse an dem Thema
• Interesse an den Büchern

Why dative? Because this is simply the case required by the phrase Interesse an.

Also remember that an is a two-way preposition, which means it can take either:

• accusative for direction/movement toward something
• dative for location or certain fixed expressions

In Interesse an, it is a fixed expression with dative.

Because German often uses no article with abstract or general nouns when speaking in a general sense.

Here Musik means music in general, so:

• Interesse an Musik = interest in music

If you said an der Musik, that would usually sound more specific, like:

• interest in the music
• interest in a particular kind of music, piece of music, or the music in a certain context

So:

• Interesse an Musik = music in general
• Interesse an der Musik = the music / that specific music

Yes, it is completely natural and common.

Interesse zeigen an means to show interest in.

You will often see it in slightly different word orders, for example:

• Sie zeigt Interesse an Musik.
• Sie zeigte großes Interesse an Musik.
• An Musik zeigt sie großes Interesse.

You may also see adjectives with Interesse:

• großes Interesse = great interest
• wenig Interesse = little interest
• echtes Interesse = genuine interest

So this sentence is standard, natural German.

Yes. That is another very common way to express the same basic idea.

Compare:

• Sie zeigt Interesse an Musik.
  = She shows interest in music.

• Sie ist an Musik interessiert.
  = She is interested in music.

The meanings are very close, but the nuance is slightly different:

• zeigt Interesse focuses on the outward display of interest
• ist interessiert describes her state or attitude more directly

Both are good German.

Yes, it is normal German main-clause word order.

The sentence is:

• Sie = subject
• zeigt = finite verb
• Interesse an Musik = object/complement

German main clauses usually place the finite verb in the second position:

• Sie zeigt Interesse an Musik.

This is often called the V2 rule (verb-second).

You could move another element to the front, but then the verb still stays second:

• An Musik zeigt sie Interesse.

That version is possible, but more marked. The original sentence is the most neutral and natural.

Interesse is a noun, and in German all nouns are capitalized.

So:

• das Interesse = the interest

That is why it begins with a capital letter.

In this sentence, Interesse is being used without an article:

• Sie zeigt Interesse an Musik.

That is normal in many set expressions and abstract contexts.

A rough English-friendly guide would be:

• Zee tsygt int-eh-RESS-uh ahn moo-ZEEK

A few important pronunciation points:

• Sie sounds like zee
• zeigt begins with ts because German z is pronounced ts
• ei in zeigt sounds like English eye
• Interesse has stress on the -res- syllable: in-te-RESS-e
• Musik is stressed on the second syllable: mu-SIK

If you want to sound more natural, keep the rhythm fairly smooth:

• Sie zeigt Interesse an Musik.

More or less, but not perfectly.

Word by word:

• Sie = she
• zeigt = shows
• Interesse = interest
• an Musik = in music / toward music, depending on how naturally you translate it

So the natural English translation is:

• She shows interest in music.

This is a good example of why direct word-for-word translation only goes so far: German uses Interesse an, while English uses interest in.