ההצלחה שלה במבחן חשובה לה מאוד, כי היא למדה כל השבוע.

In Hebrew, present-tense sentences like this often do not use a separate verb meaning to be.

So:

ההצלחה שלה במבחן חשובה לה מאוד

literally works like:

Her success in the test very important to her.

That is normal Hebrew.
If you wanted past or future, then Hebrew would usually use a form of היה:

• ההצלחה שלה הייתה חשובה לה = Her success was important to her
• ההצלחה שלה תהיה חשובה לה = Her success will be important to her

ההצלחה שלה means her success in a specific, definite sense.

The ה at the beginning makes הצלחה definite: the success.
So ההצלחה שלה is basically her success / the success she had.

If you said הצלחה שלה, that would sound more like:

• a success of hers
• one success she had

So the version in the sentence points to a particular success, not just any success.

That is the normal Hebrew pattern for this kind of possession.

Hebrew commonly says:

• הספר שלה = her book
• הבית שלו = his house
• החברה שלהם = their company

Literally, it is more like the book of hers, the house of his, and so on.

English puts the possessor first (her success), but Hebrew usually puts it after the noun in everyday speech.

Yes. הצלחתה במבחן also means her success in the exam.

The difference is mostly style:

• ההצלחה שלה = very common, natural, everyday modern Hebrew
• הצלחתה = also correct, but often feels a bit more formal, written, or compact

So both are good Hebrew, but the sentence you were given uses the more conversational pattern.

במבחן is made from:

• ב = in / on / at
• המבחן = the test / the exam

When ב comes before ה, they combine:

• ב + המבחן → במבחן

This is very common in Hebrew.

So במבחן means something like:

• in the exam
• on the test

English choice depends on context, but the Hebrew form is normal.

Because it agrees with ההצלחה, and הצלחה is a feminine singular noun.

In Hebrew, adjectives must agree with the noun they describe in gender and number.

So:

• הצלחה = feminine singular
• therefore חשובה = feminine singular

It does not agree with לה (to her).
It agrees with the subject: the success.

Compare:

• ההצלחה חשובה = The success is important
• ההישג חשוב = The achievement is important
  because הישג is masculine

לה means to her.

With חשוב / חשובה (important), Hebrew usually says who something is important to by using ל־:

• חשוב לי = important to me
• חשוב לך = important to you
• חשובה לה = important to her

So:

ההצלחה שלה במבחן חשובה לה מאוד
= Her success in the exam is very important to her

This ל־ structure is very common with feelings, relevance, and importance.

מאוד means very.

In this sentence, חשובה לה מאוד is a very natural word order:
important to her מאוד = very important to her

Hebrew is somewhat flexible here. You could also hear:

• מאוד חשובה לה
• חשובה מאוד לה (less neutral in many contexts)

The original version sounds natural and standard.

That is a great question.
Yes, למדה already tells you the subject is she (third person feminine singular).

So Hebrew could also say:

כי למדה כל השבוע

However, adding היא is very natural when you want to make the subject explicit or clearer.

Here it helps because the previous clause had ההצלחה as the grammatical subject.
By saying כי היא למדה, the sentence clearly shifts back to the woman herself:

• she studied all week

So היא is not strictly necessary, but it improves clarity.

No, they are different.

• כל השבוע = all week / the whole week
• כל שבוע = every week

The ה matters.

So in your sentence:

היא למדה כל השבוע
means she studied during the entire week leading up to the test.

It does not mean that she studies every week in general.

Yes, to some extent.

For example, these are all possible:

• ההצלחה שלה במבחן חשובה לה מאוד, כי היא למדה כל השבוע.
• ההצלחה שלה במבחן מאוד חשובה לה, כי היא למדה כל השבוע.
• הצלחתה במבחן חשובה לה מאוד, כי היא למדה כל השבוע.

They all express basically the same idea, but with small differences in style or emphasis.

The original sentence is perfectly natural and clear.