Que me regañen delante de toda la clase hace que me ponga nerviosa.

Because que here is a conjunction, not a question word.

• que = that, or it introduces a subordinate clause
• qué = what? in direct or indirect questions, or in exclamations

So Que me regañen... is not asking what; it is introducing the idea that they scold me / being scolded.

Because que me regañen delante de toda la clase is functioning as a whole clause that acts like the subject of the sentence.

In English, we often prefer something like:

• Being told off in front of the whole class makes me nervous
• The fact that they tell me off in front of the whole class makes me nervous

Spanish often uses que + verb to build that kind of subject clause.

Because Spanish is treating this as an event or situation, not simply as a neutral factual statement.

Here, que me regañen... is more like:

• being told off...
• the idea/situation of them telling me off...

That is why the subjunctive regañen sounds natural. If you used regañan, it would sound more like a straightforward statement of fact or habit, which is less natural in this structure.

The verb regañen is third person plural, and Spanish often leaves the subject unstated.

So the meaning is something like:

• when they scold me
• when people scold me
• when I get told off

The exact people are not important here; the sentence leaves them unspecified.

They do two different jobs.

• In me regañen, me is the object of regañar: they scold me
• In me ponga nerviosa, me belongs to ponerse: to become/get

So:

• me regañen = they tell me off
• me ponga nerviosa = I get nervous / I become nervous

Same pronoun form, but different grammatical roles.

Here hace que means makes or causes.

So:

• hace que me ponga nerviosa = makes me get nervous / causes me to become nervous

It is a very common Spanish pattern:

• hacer que + subjunctive

It expresses that one thing causes another thing to happen.

Because after hacer que, Spanish normally uses the subjunctive.

So:

• hace que me ponga nerviosa = correct
• hace que me pongo nerviosa = not correct here

This is because the second verb is presented as the result caused by the first situation.

Because the adjective agrees with the person speaking.

• A female speaker says nerviosa
• A male speaker says nervioso

So this sentence suggests that the speaker is female.

Delante de is a fixed expression meaning in front of or before.

• delante de la clase = in front of the class
• delante de toda la clase = in front of the whole class

And toda agrees with clase, which is a singular feminine noun.

So toda la clase means the entire class / everyone in the class.

Yes. That is also a very natural sentence.

Compare:

• Que me regañen delante de toda la clase hace que me ponga nerviosa
• Me pone nerviosa que me regañen delante de toda la clase

The second version is a bit more direct and common in everyday speech. The original version gives more emphasis to the situation itself first.

Not regañarme, because regañarme means to scold myself.

If you want an infinitive idea, Spanish would need something like:

• Ser regañada delante de toda la clase me pone nerviosa

That is grammatical, but it sounds more formal and less natural in everyday conversation. Que me regañen... is more idiomatic here.

Because ponerse often means to become or to get, especially with emotional or physical states.

So:

• me pongo nerviosa = I get nervous / I become nervous
• estoy nerviosa = I am nervous

In this sentence, the idea is that the situation causes a change of state, so ponerse fits very well.