Marie veut corriger ses fautes elle-même.

Because ses is the possessive adjective used with a plural noun.

• faute = mistake, fault
• fautes = mistakes, faults

So:

• sa faute = her mistake / her fault
• ses fautes = her mistakes / her faults

In French, possessive adjectives agree with the thing owned, not with the owner. Since fautes is plural, French uses ses.

French possessive adjectives do not show the gender of the owner. They show the number and sometimes the gender of the noun that follows.

So ses fautes can mean:

• his mistakes
• her mistakes

You only know it is her here because the subject is Marie.

In French, after many verbs, a second verb can come directly in the infinitive.

• veut corriger = wants to correct

French does not need a separate word equivalent to English to here.

This is very common:

• Je veux partir. = I want to leave.
• Elle aime lire. = She likes reading / likes to read.

So Marie veut corriger... is completely normal.

Because it depends on the conjugated verb veut.

In Marie veut corriger ses fautes, the main conjugated verb is:

• veut = wants

The next verb stays in the infinitive:

• corriger = to correct

This is the standard pattern after vouloir:

• Je veux manger.
• Nous voulons apprendre.
• Marie veut corriger ses fautes.

elle-même means herself or by herself / herself personally, depending on context.

In this sentence, it adds emphasis:

• Marie veut corriger ses fautes. = Marie wants to correct her mistakes.
• Marie veut corriger ses fautes elle-même. = Marie wants to correct her mistakes herself.

It suggests that Marie wants to do it personally, rather than having someone else do it for her.

At the end, elle-même strongly emphasizes Marie as the person doing the action.

So the focus is:

• Marie wants to correct her own mistakes herself

This end position is very natural in French for emphasis.

You may also see other placements in different contexts, but here the final position sounds clear and idiomatic.

Because stressed pronouns with -même are written with a hyphen in French.

Examples:

• moi-même = myself
• toi-même = yourself
• lui-même = himself
• elle-même = herself
• nous-mêmes = ourselves

So elle-même is just the standard spelling.

Not exactly. They are close in some contexts, but they are not the same.

• elle-même emphasizes that she herself is the one doing it
• seule emphasizes that she is alone

Compare:

• Marie veut corriger ses fautes elle-même.
  = Marie wants to correct her mistakes herself.

• Marie veut corriger ses fautes seule.
  = Marie wants to correct her mistakes alone.

Sometimes both ideas overlap, but the meanings are different.

It can definitely mean mistakes, and that is often the most natural translation here.

faute is a flexible word. Depending on context, it can mean:

• mistake
• error
• fault
• wrong

In a learning or writing context, corriger ses fautes usually means correct her mistakes.

Yes, that is possible, but the emphasis changes slightly.

• Marie veut corriger ses fautes elle-même.
  Strong natural emphasis on doing it herself

• Marie veut elle-même corriger ses fautes.
  Slightly more formal or stylistic; it emphasizes Marie herself as the one who wants to do it

Both are grammatical, but the original sentence is probably the most natural for everyday use.

A learner might notice a few useful points:

• veut sounds roughly like vuh
• corriger has a soft g sound: ko-ree-zhay
• ses sounds like say
• fautes sounds like foht
• elle-même sounds roughly like el-mem

Also, French rhythm links words together smoothly. A rough pronunciation guide for the whole part is:

vuh ko-ree-zhay say foht el-mem

That is only approximate, but it helps an English speaker get started.

No. It is an emphatic / stressed pronoun expression, not a reflexive pronoun.

The reflexive pronoun for she would be se, as in:

• Elle se corrige. = She corrects herself.

But in your sentence:

• elle-même adds emphasis
• it does not make the verb reflexive

So Marie veut corriger ses fautes elle-même means Marie wants to correct her mistakes herself, not necessarily that she is correcting herself in the grammatical sense.