Me pongo nervioso cuando tengo que hablar en público.

Both can be correct, but they mean slightly different things.

• Me pongo nervioso = I get nervous / I become nervous
  • It focuses on the change of state.
• Estoy nervioso = I am nervous
  • It describes the state itself.

So in this sentence, me pongo nervioso cuando... means I get nervous when..., which sounds natural because the nervousness starts in that situation.

Here, ponerse means to become in a temporary or emotional/physical state.

Common pattern:

• ponerse + adjective

Examples:

• ponerse nervioso = to get nervous
• ponerse triste = to get sad
• ponerse contento = to become happy
• ponerse enfermo = to get sick

So me pongo nervioso literally works like I put myself nervous, but in real English it means I get nervous.

Because ponerse is a reflexive verb.

Its forms are:

• me pongo = I get
• te pones = you get
• se pone = he/she gets
• nos ponemos = we get
• os ponéis = you all get
• se ponen = they/you all get

The me is the reflexive pronoun that matches yo.

So:

• me pongo nervioso = I get nervous

Because nervioso is an adjective, and after ponerse, Spanish normally uses an adjective to describe the new state.

• nervioso = nervous
• nerviosamente = nervously

Compare:

• Me pongo nervioso = I get nervous
• Hablo nerviosamente = I speak nervously

In your sentence, the speaker is describing how they become, so the adjective nervioso is the right choice.

Yes. The adjective must agree with the speaker.

• A man would usually say: Me pongo nervioso
• A woman would usually say: Me pongo nerviosa

This is because nervioso/nerviosa agrees in gender with the person speaking.

Because this sentence describes something that generally happens or usually happens.

• cuando tengo que hablar en público = when I have to speak in public

Spanish uses the present indicative after cuando when talking about habitual or repeated situations.

Examples:

• Me pongo nervioso cuando hablo con desconocidos.
• Me siento mejor cuando descanso.

If it referred to a future event, Spanish often uses the present subjunctive after cuando:

• Me pondré nervioso cuando tenga que hablar en público mañana.
  • I’ll get nervous when I have to speak in public tomorrow.

Tener que + infinitive means to have to do something.

So:

• tengo que hablar = I have to speak
• tienes que estudiar = you have to study
• tenemos que salir = we have to leave

In your sentence:

• cuando tengo que hablar en público = when I have to speak in public

This is one of the most common ways to express obligation in Spanish.

Because en público is a fixed expression meaning in public / publicly / before an audience.

• hablar en público = to speak in public
• estar en público = to be in public

You normally do not use the article here.

Compare:

• Hablo en público = I speak in public
• Estoy en público = I’m in public

But el público means the audience / the public:

• El público aplaudió. = The audience applauded.

So:

• hablar en público = speak publicly / before people
• hablar al público = speak to the audience

Those are related, but not the same.

Yes, but the meaning shifts slightly.

• hablar en público = to speak in public / do public speaking
• hablar delante de mucha gente = to speak in front of many people

The first is more like the general idea of public speaking.
The second emphasizes being physically in front of lots of people.

So both can work, but en público is the most natural phrase for public speaking.

Because Spanish often leaves out subject pronouns when the verb already shows who the subject is.

• Me pongo nervioso already tells you the subject is yo because pongo is the first person singular form.

So:

• Me pongo nervioso = normal
• Yo me pongo nervioso = also correct, but more emphatic

You might add yo if you want contrast:

• Yo me pongo nervioso, pero mi hermano no.
  • I get nervous, but my brother doesn’t.

Usually, no. In this kind of sentence, ponerse nervioso is understood as to become nervous, not to intentionally make oneself nervous.

So:

• Me pongo nervioso cuando... = I get nervous when...

If you wanted to say that something makes you nervous, Spanish often uses:

• Me pone nervioso... = It makes me nervous...

For example:

• Me pone nervioso hablar en público.
  • Speaking in public makes me nervous.

Notice the difference:

• Me pongo nervioso = I get nervous
• Me pone nervioso = It makes me nervous

The sentence breaks down like this:

• Me pongo nervioso = I get nervous
• cuando = when
• tengo que hablar = I have to speak
• en público = in public

So the structure is:

[main clause] + cuando + [subordinate clause]

More literally:

• I get nervous when I have to speak in public.

It’s a very common Spanish pattern:

• Me enfado cuando llegas tarde. = I get annoyed when you arrive late.
• Me alegro cuando vienes. = I’m happy when you come.