De docent past de les aan het niveau van elke leerling aan.

They come from two different sources:

• The first aan is the separable prefix of the verb aanpassen (“to adapt”). In a main clause, that prefix gets pulled off and moved to the end.
• The second aan is the preposition meaning “to” or “according to,” introducing het niveau van elke leerling.

They just happen to look identical in spelling but serve different grammatical roles.

Aanpassen is a separable verb: the base is passen, the prefix is aan. To conjugate in the present tense:

• ik pas aan
• jij past aan
• hij/zij/het past aan
• wij passen aan
  … and so on.

In our sentence, the subject is de docent (third-person singular), so you use past. Because it’s a main clause, the prefix aan goes to the very end, yielding past … aan.

Dutch word order for separable verbs in a main clause is:

1. Subject (de docent)
2. Finite verb (past)
3. Other elements (object, adverbials)
4. Separable prefix (aan)

So de les (the direct object) naturally sits after past and before the moved-prefix aan.

Yes, you can say iedere leerling or elke leerling with almost the same meaning (“each/every pupil”).

• elke often stresses each individual item separately.
• iedere can sound slightly more formal or general.

But in everyday use they’re interchangeable here.

Since les is a de-word, you can use the object pronoun hem:

• De docent past hem aan.

You could also use a demonstrative pronoun if you’ve just mentioned the lesson:

• De docent past die aan.

Yes. A common synonym is afstemmen op, also a separable verb:

• De docent stemt de les af op het niveau van elke leerling.
  Here op is the preposition, and in a main clause op moves to the end in the same way.