Sie legt ihren Gürtel auf den Stuhl.

Here, Sie means she.

At the start of a German sentence, sie and Sie both look the same because the first word is always capitalized. The clue is the verb:

• sie legt = she puts / she lays
• sie legen = they put / they lay
• Sie legen = you put / you lay (formal)

Because the sentence has legt, not legen, it must be she.

Legt is the 3rd person singular present tense form of legen.

The subject is Sie = she, so the verb has to match that subject:

• ich lege = I put
• du legst = you put
• er/sie/es legt = he/she/it puts
• wir/sie legen = we/they put

So Sie legt means she puts or she is putting.

This is a very common German distinction:

• legen = to lay / to put something somewhere
  
  • it involves an action
  • it usually takes a direct object
• liegen = to lie / to be lying somewhere
  
  • it describes a position or state

So:

• Sie legt ihren Gürtel auf den Stuhl. = She puts her belt onto the chair.
• Ihr Gürtel liegt auf dem Stuhl. = Her belt is lying on the chair.

English often uses lie and lay less carefully in everyday speech, but German keeps this distinction very clearly.

Because Gürtel is the direct object, and it is masculine singular.

• der Gürtel = the belt
• masculine singular in the accusative becomes den Gürtel
• the possessive ihr- has to match that case and gender, so it becomes ihren

So:

• ihr Gürtel = her belt as the subject
• ihren Gürtel = her belt as the object

That is why the sentence says Sie legt ihren Gürtel ...

Yes. In this sentence, ihren means her.

It comes from the possessive word ihr-, which matches sie = she.

So:

• Sie = she
• ihren Gürtel = her belt

German possessives change their endings depending on case, gender, and number, so her does not always appear as plain ihr.

Because ihren already does the job that an article would normally do.

German usually uses:

• den Gürtel = the belt
• einen Gürtel = a belt
• ihren Gürtel = her belt

You do not combine them in normal German. So you do not say something like ihren den Gürtel.

This works much like English:

• the belt
• a belt
• her belt

Because auf is a two-way preposition.

With two-way prepositions, German uses:

• accusative for movement toward a destination
• dative for location with no movement

In this sentence, the belt is being moved onto the chair, so German uses the accusative:

• auf den Stuhl = onto the chair

If the belt were already there, you would use the dative:

• auf dem Stuhl = on the chair

So compare:

• Sie legt ihren Gürtel auf den Stuhl. = She puts her belt onto the chair.
• Der Gürtel liegt auf dem Stuhl. = The belt is on the chair.

Because Stuhl is a masculine noun:

• der Stuhl = the chair

After auf with movement, German needs the accusative. The masculine accusative article of der is den:

• nominative: der Stuhl
• accusative: den Stuhl
• dative: dem Stuhl

So auf den Stuhl is exactly what you expect for movement onto a masculine noun.

No, German word order is somewhat flexible, but this version is the most neutral and natural:

• Sie legt ihren Gürtel auf den Stuhl.

This is basically:

• subject: Sie
• verb: legt
• object: ihren Gürtel
• place/destination: auf den Stuhl

You can move parts around for emphasis, as long as the verb stays in the second position in a main clause:

• Ihren Gürtel legt sie auf den Stuhl.
• Auf den Stuhl legt sie ihren Gürtel.

Those alternatives are possible, but they sound more marked or more emphatic than the original.

Because all German nouns are capitalized.

So:

• der Gürtel
• der Stuhl

This is one of the most noticeable spelling rules in German. It helps learners spot nouns more easily.