Je n’ai pas fait exprès de te réveiller.

Faire exprès is a fixed expression meaning to do something on purpose / deliberately.

So in the negative:

Je n’ai pas fait exprès...
means I didn’t do it on purpose...

It is best learned as a whole expression, because the literal pieces do not help much.

This is the passé composé of faire:

• j’ai fait = I did
• je n’ai pas fait = I did not do

French forms the negative around the auxiliary verb:

• je n’ai pas fait
• not je ai ne pas fait

Also, ne becomes n’ before a vowel sound, which is why you get n’ai.

After faire exprès, French commonly uses de + infinitive to say what the intentional action was.

So:

• faire exprès de + infinitive

Examples:

• Il a fait exprès de partir tôt.
• Je n’ai pas fait exprès de te réveiller.

Here, de te réveiller specifies the action: to wake you up.

If the action is already obvious, French can also simply say:

• Je n’ai pas fait exprès.

Because after de, the verb is in the infinitive:

• de réveiller = to wake
• de te réveiller = to wake you

So the sentence structure is not using a second conjugated verb.
It is:

• I did not do [it] on purpose to wake you

That is why French uses the infinitive réveiller, not the past participle réveillé.

Because object pronouns in French go before the infinitive they belong to.

Here, te is the object of réveiller:

• réveiller quelqu’un = to wake someone
• te réveiller = to wake you

So French says:

• de te réveiller

not:

• de réveiller te

No. Here te means you, and it is the direct object of réveiller.

• réveiller quelqu’un = to wake someone
• se réveiller = to wake up

In this sentence, the subject is je and the object is te, so they are different people:

• I woke you

That means this is not reflexive.

The main verb is in the passé composé, which is a very common French past tense.

• j’ai fait = I did
• je n’ai pas fait = I didn’t do

The infinitive réveiller does not have its own tense here; it is just naming the action.

So the whole sentence is talking about a past situation: you woke someone up, but not intentionally.

Fait is the past participle of faire, and here it stays fait.

In this sentence, there is no preceding direct object that would make it agree, so it remains unchanged.

Also, exprès never agrees either. It always stays exprès.

So:

• Je n’ai pas fait exprès
• Elle n’a pas fait exprès
• Ils n’ont pas fait exprès

All keep fait exprès the same.

No. Exprès is invariable.

That means it does not change for masculine, feminine, singular, or plural.

Examples:

• Il l’a fait exprès.
• Elle l’a fait exprès.
• Ils l’ont fait exprès.

The form exprès stays the same every time.

Yes. That is also a very natural sentence.

It means essentially the same thing:
I didn’t wake you up on purpose.

The difference is mostly one of structure:

• Je n’ai pas fait exprès de te réveiller.
  = I didn’t do it on purpose, namely wake you.

• Je ne t’ai pas réveillé exprès.
  = I didn’t wake you up on purpose.

The second version is often very direct and common in conversation.

Yes. It sounds natural, especially when apologizing or explaining that something happened accidentally.

For example, if someone says:

• Tu m’as réveillé ! = You woke me up!

You might reply:

• Désolé, je n’ai pas fait exprès de te réveiller.

It is a normal way to say that the action was unintentional.

Yes, very often.

If the context is already clear, French speakers often stop at:

• Je n’ai pas fait exprès.

That simply means:

• I didn’t do it on purpose.

The longer version adds the specific action:

• Je n’ai pas fait exprès de te réveiller.

So the short version is common when everyone already knows what happened.