Anna slaagt erin om elke week tijd vrij te maken voor haar studie.

Here slagen does not mean just to pass an exam.
The combination erin slagen means to succeed in doing something or to manage to do something.

So:

• Anna slaagt erin ... = Anna succeeds in / manages to ...

On its own, slagen often means:

• to pass: Ze slaagt voor het examen = she passes the exam
• to turn out well: Het feest is goed geslaagd = the party turned out well

So erin is an important part of the meaning in this sentence.

Because slagen normally goes with the preposition in when it means to succeed in:

• slagen in iets = to succeed in something

When Dutch refers not to a noun but to a whole action or clause, it often uses an er + preposition form.
That is why in becomes erin.

Compare:

• Anna slaagt in haar opdracht. = Anna succeeds in her assignment.
• Anna slaagt erin om tijd vrij te maken. = Anna succeeds in making time available.

So erin is basically the Dutch way of saying something like in that / in doing that.

Not always. You will commonly see both patterns:

• erin slagen iets te doen
• erin slagen om iets te doen

So this sentence is fine, but you may also see:

• Anna slaagt erin elke week tijd vrij te maken voor haar studie.

In other words, om introduces the infinitive clause, but Dutch speakers do not always include it after erin slagen.

As a learner, the safest thing is:

• recognize both,
• and don’t be surprised if Dutch texts use either version.

Because vrijmaken is a separable verb.

Its parts are:

• vrij = free
• maken = make

When a separable verb is used with te + infinitive, te goes between the two parts:

• vrijmaken → vrij te maken
• opbellen → op te bellen
• aandoen → aan te doen

So:

• tijd vrijmaken = to free up time
• tijd vrij te maken = to free up time

That split is completely normal.

Tijd vrijmaken means to free up time, to make time available, or to set aside time.

It is the normal Dutch expression for the idea of intentionally finding time for something.

• Ik moet tijd vrijmaken voor mijn werk.
• Kun je morgen tijd vrijmaken?

Just tijd maken is usually not the natural Dutch expression for this meaning.
A native speaker would much more likely say:

• tijd vrijmaken
• tijd hebben
• tijd vinden

So for English make time, Dutch usually prefers tijd vrijmaken.

Yes, elke week and iedere week are both possible here.

They both mean every week.

In many everyday contexts, the difference is very small or nonexistent.
Often:

• elke week sounds slightly more common in casual speech
• iedere week can sound a little more emphatic, like each and every week

But in this sentence, either would work:

• Anna slaagt erin om elke week ...
• Anna slaagt erin om iedere week ...

Because voor fits the meaning of for the benefit/purpose of her studies.

• tijd vrijmaken voor haar studie = free up time for her studies

If you used aan, you would usually need a different verb or structure, for example:

• tijd besteden aan haar studie = spend time on her studies
• werken aan haar studie = work on her studies

So:

• vrijmaken voor = make available for
• besteden aan = spend on

Both are correct Dutch, but they belong to different expressions.

In Dutch, studie is often singular where English might think in a broader way, like study, course of study, or education.

So:

• haar studie can mean her studies in general, her degree program, or the studying she has to do

Dutch very often uses the singular here:

• Ik ben druk met mijn studie.
• Zij heeft weinig tijd voor haar studie.

Studies does exist, but it is less natural in many everyday cases and may suggest:

• multiple separate studies,
• a more formal style,
• or a slightly different nuance depending on region.

So haar studie is the most natural choice here.

No. Haar here refers to Anna, not to studie.

Dutch possessive words such as:

• mijn
• jouw
• zijn
• haar
• ons
• hun

agree with the owner, not with the noun that follows.

So because Anna is female:

• haar studie = her studies / her course of study

Even though studie is a de-word, that does not affect the choice of haar.

This is an infinitive clause introduced by om.

A helpful way to see it is:

• om = introduces the clause
• elke week = time expression
• tijd = object
• vrij te maken = separable infinitive with te
• voor haar studie = prepositional phrase

Dutch often puts the verbal part toward the end of the clause, and with separable verbs the parts stay split:

• tijd vrij te maken

So the order is quite natural:

• om elke week tijd vrij te maken voor haar studie

You may also see other acceptable orders, for example:

• om voor haar studie elke week tijd vrij te maken

But the original sentence sounds very normal.

Slaagt is the third person singular present tense of slagen.

The subject is Anna, so Dutch uses:

• ik slaag
• jij slaagt / slaag je
• hij/zij slaagt
• wij slagen
• jullie slagen
• zij slagen

So:

• Anna slaagt erin ... = Anna succeeds / manages ...

If you changed the subject, the verb would change too:

• Ik slaag erin ...
• Wij slagen erin ...