Siento orgullo cuando mi hija habla español con tanta naturalidad.

Both are possible, but they are not exactly the same.

• Siento orgullo means I feel pride or I feel proud in an emotional sense.
• Estoy orgulloso / Estoy orgullosa means I am proud and uses an adjective.

So:

• Siento orgullo cuando... focuses on the feeling itself.
• Estoy orgulloso/a de que... focuses more on your state or attitude.

Also, orgulloso/a must agree with the speaker’s gender, but orgullo does not change:

• a man: Estoy orgulloso
• a woman: Estoy orgullosa

Spanish often leaves out subject pronouns when they are clear from the verb ending.

• Siento already means I feel
• so yo siento is usually unnecessary

You can add yo for emphasis or contrast, but normally it is omitted:

• Siento orgullo... = normal
• Yo siento orgullo... = more emphatic, like I feel proud

Because this sentence is describing something that really happens or is seen as a fact, not a wish, doubt, or emotion introduced by a structure that requires the subjunctive.

• cuando mi hija habla español... = when my daughter speaks Spanish...

Here, cuando is followed by the present indicative because it refers to a habitual or real situation.

Compare:

• Siento orgullo cuando mi hija habla español... = this happens, and I feel proud when it does
• Sentiré orgullo cuando mi hija hable español... = I will feel proud when my daughter speaks Spanish someday in the future

So with future or not-yet-real events, Spanish often uses the subjunctive after cuando.

Because the sentence expresses a repeated or general situation in the present.

• cuando mi hija habla español means when my daughter speaks Spanish
• it suggests something that happens from time to time, or whenever that situation occurs

This is very common in Spanish:

• Me alegro cuando vienes = I’m happy when you come
• Sonríe cuando escucha esa canción = She smiles when she hears that song

If the event were in the future and not yet happened, Spanish would usually use the subjunctive:

• Me alegraré cuando vengas

Because naturalidad is a noun, and tanto/a/os/as is used before nouns.

• tanta naturalidad = so much naturalness / such naturalness
• tan is used before adjectives or adverbs:
  • tan natural = so natural
  • tan bien = so well

So:

• con tanta naturalidad = with such naturalness
• de forma tan natural = in such a natural way

Both are good Spanish, but they are built differently.

Naturalidad means something like naturalness, ease, or effortlessness.

In this sentence, it suggests that the daughter speaks Spanish:

• comfortably
• smoothly
• without sounding forced

In natural English, you would often translate it less literally:

• with such ease
• so naturally

So con tanta naturalidad does not just mean in a natural way mechanically; it implies fluency and comfort.

Because when talking about a language after verbs like hablar, aprender, estudiar, or enseñar, Spanish usually does not use the article.

• habla español = she speaks Spanish
• aprendo francés = I’m learning French
• estudia inglés = he studies English

Using el español would usually sound more like the Spanish language in a more specific or formal sense.

So here, habla español is the normal everyday structure.

Yes, but it changes the structure.

Two common patterns are:

• Siento orgullo cuando mi hija habla español...
• Siento orgullo de que mi hija hable español...

The first means:

• I feel proud when my daughter speaks Spanish...

The second means:

• I feel proud that my daughter speaks Spanish...

With de que, the verb after it normally goes in the subjunctive:

• de que mi hija hable

So both are correct, but they express the idea in different ways.

In this sentence, mi hija is needed because it is the subject of habla.

• cuando mi hija habla español = when my daughter speaks Spanish

If you removed mi hija, the sentence would become:

• Siento orgullo cuando habla español

That could still be grammatical, but now it would mean:

• I feel proud when he/she speaks Spanish

The listener would need context to know who habla refers to. So mi hija is included for clarity.

This is a very normal Spanish word order:

• Siento orgullo = main clause
• cuando mi hija habla español con tanta naturalidad = time clause

Spanish word order is fairly flexible, so you could also say:

• Cuando mi hija habla español con tanta naturalidad, siento orgullo.

That version puts more emphasis on the situation first. Both are correct.

The original order sounds natural and straightforward.

Yes. Sentir orgullo is a common expression, and orgullo often appears without an article in this structure.

• Siento orgullo = I feel pride
• Sintió orgullo por su hijo = He/She felt pride for his/her son

You may also see:

• Siento un gran orgullo... = I feel great pride...
• Siento mucho orgullo... = I feel a lot of pride...

So no article is perfectly normal here.

No. Spanish offers several natural alternatives, with slightly different nuances:

• Siento orgullo cuando mi hija habla español con tanta naturalidad.
• Siento orgullo cuando mi hija habla español tan naturalmente.
• Siento orgullo cuando mi hija habla español de forma tan natural.

The original sentence is especially elegant because con tanta naturalidad sounds very idiomatic and expressive in Spanish.

A learner should remember this pattern:

• con + tanta/o/os/as + noun
• for example: con tanta fuerza, con tanta calma, con tanta seguridad

Yes, it sounds completely natural in Spain.

It is:

• grammatically correct
• idiomatic
• emotionally natural

A speaker from Spain might also say:

• Me siento orgulloso/a cuando mi hija habla español con tanta naturalidad.
• Estoy muy orgulloso/a cuando mi hija habla español así de bien.

But the original sentence is already very good Spanish.