Todavía me da miedo hablar en público.

Both can express fear, but they are built differently.

• Tener miedo = to be afraid / to have fear
  • Tengo miedo de hablar en público.
• Dar miedo = to give someone fear / to scare someone
  • Me da miedo hablar en público.

In me da miedo hablar en público, the idea is literally closer to:

• Speaking in public gives me fear

This structure is very common in Spanish. It often sounds natural when talking about things or actions that cause fear.

Me is an indirect object pronoun, meaning to me.

So:

• me da miedo = it gives fear to me

Other forms would be:

• te da miedo = it scares you
• le da miedo = it scares him/her/you (formal)
• nos da miedo = it scares us

In this sentence, me shows who feels the fear.

The subject is hablar en público.

So the structure is:

• Hablar en público = speaking in public
• me da miedo = gives me fear

Literally:

• Speaking in public gives me fear

That is why the verb is da and not dan: the whole idea hablar en público is treated as one singular action.

In Spanish, an infinitive can act like a noun-like idea, just as speaking does in English.

So:

• hablar en público = speaking in public

Spanish often uses the infinitive where English uses the -ing form.

Examples:

• Fumar es malo. = Smoking is bad.
• Viajar me encanta. = I love travelling.
• Hablar en público me da miedo. = Speaking in public scares me.

Here todavía means still.

So it adds the idea that this feeling continues now:

• Todavía me da miedo... = It still scares me...

It suggests that maybe the speaker expected this fear to disappear, but it has not.

A very similar word is aún:

• Aún me da miedo hablar en público.

In this sentence, todavía and aún are basically interchangeable.

Yes, sometimes the position can change, although the beginning is very natural here.

Common options:

• Todavía me da miedo hablar en público.
• Me da miedo todavía hablar en público. (possible, but less natural in many contexts)
• Me da todavía miedo hablar en público. (possible, but less common)

For a learner, the safest and most natural option is:

• Todavía me da miedo hablar en público.

En público is a fixed expression meaning:

• in public
• before an audience
• publicly

So:

• hablar en público = to speak in public / to speak before an audience

By contrast, al público usually means to the public:

• Habló al público = He spoke to the public / audience

So the meanings are different:

• hablar en público = speak publicly
• hablar al público = address the public/audience

Because en público is an established expression.

Just as Spanish says:

• en casa = at home
• en clase = in class
• en público = in public

You normally do not say en el público here.

El público exists, but it means the audience / the public as a noun:

• El público aplaudió. = The audience applauded.

So in your sentence, público is part of the expression en público, not a separate noun with an article.

Yes, absolutely. That is also correct and very natural.

Compare:

• Todavía me da miedo hablar en público.
• Todavía tengo miedo de hablar en público.

The difference is mainly in structure:

• me da miedo focuses on the thing that causes fear
• tengo miedo de focuses on the person experiencing fear

Both are common, and both would sound normal in Spain.

Because dar miedo and tener miedo use different patterns.

With tener miedo, you often say:

• Tengo miedo de hablar en público.

With dar miedo, you usually say:

• Me da miedo hablar en público.

So:

• tener miedo de + infinitive
• dar miedo + infinitive

That is why de appears with tener miedo, but not normally with dar miedo in this kind of sentence.

Yes. It sounds completely natural in Spain.

A speaker from Spain might also say similar things like:

• Todavía me pone nervioso hablar en público.
• Aún me da miedo hablar en público.
• Sigo teniendo miedo de hablar en público.

But your original sentence is standard, natural, and easy to understand everywhere in the Spanish-speaking world.