Learn Hardcore Portuguese (Portugal): Quando a trovoada começa, eu vou para dentro de casa. - When the thunderstorm starts, I go inside the house.

Trovoada is usually best translated as thunderstorm. It refers to a storm that involves thunder (and typically lightning and rain too), not just the sound of thunder on its own.
Related words:

trovão – thunder, a thunderclap
relâmpago – a flash of lightning (what you see)
raio – a lightning bolt, lightning strike
tempestade – a storm in general (can be wind, rain, etc., not necessarily with thunder)

So há trovoada would be something like there is a thunderstorm.

Why do we say a trovoada with the article a?

In European Portuguese, natural phenomena and weather events very often take the definite article:

a chuva (the rain)
o vento (the wind)
a trovoada (the thunderstorm)

The article a here is not pointing to one specific, previously mentioned storm in the way English the does; it is just standard usage.
You could sometimes omit the article in a more general statement, for example trovoada assusta-me (thunderstorms scare me), but a trovoada is the default, natural choice in most contexts like this sentence.

How do you pronounce trovoada and dentro in European Portuguese?

Very roughly, in English-like spelling:

trovoada ≈ troh-voo-AH-da
- 4 syllables: tro-vo-a-da
- Main stress on -a- (the -a-da part).
dentro ≈ DEN-troo
- The en is nasal (similar to the French sound in bien), and the final o sounds more like u.

In IPA for European Portuguese, a common pronunciation is roughly:

trovoada – [tɾovuˈaðɐ]
dentro – [ˈdẽtɾu]

Do I have to say eu, or could I just say vou para dentro de casa?

You can absolutely drop eu here.
Portuguese is a pro‑drop language: because the verb form vou already tells you the subject is eu, it is very natural to say:

Quando a trovoada começa, vou para dentro de casa.

Adding eu is possible but usually only done:

for emphasis (eu vou, not someone else), or
in contexts where you especially want to make the subject clear.

In everyday speech, most people would omit eu in this sentence.

What is the difference between para dentro de casa, para casa, em casa, and dentro de casa?

They all involve the idea of home/house, but they are not the same:

para dentro de casa – literally to inside of the house.
Emphasises the movement across a boundary from outside to inside the building.
para casa – (going) home, focusing more on the destination (home) than on the outside/inside contrast.
Example: Vou para casa às seis (I go home at six).
em casa – at home / in the house, location only, no movement.
Example: Estou em casa (I am at home).
dentro de casa – inside the house, location inside, again without the idea of moving there.
Example: As crianças estão dentro de casa (The children are inside the house).

Your sentence uses vou para dentro de casa because it describes a movement into the interior of the house.

Why is it de casa and not da casa?

De casa here has a more general or abstract meaning of home / the house as my place, without focusing on a specific, identified house. In this kind of expression:

dentro de casa
em casa
chegar a casa / de casa

the word casa behaves almost like home in English, and often appears without a definite article.

If you said dentro da casa, that would sound like you are talking about one specific house that both speakers have in mind, for example a particular building you have been describing.

Why do we say vou para dentro de casa and not vou dentro de casa?

In Portuguese, movement towards a place usually needs a preposition that expresses direction:

vou para dentro (I go to the inside)
vou para casa (I go home)
vou ao jardim (I go to the garden)

Dentro on its own is mostly used for location (being inside), not for movement:

Estou dentro de casa (I am inside the house).

So you say:

vou para dentro de casa (movement into the house),
not vou dentro de casa, which sounds wrong to native speakers.

Why is it quando a trovoada começa (present tense) and not quando a trovoada começar?

Both forms are possible, but they are not identical in nuance:

Quando a trovoada começa, eu vou para dentro de casa.
Present indicative in both clauses; this sounds like a habit or a general rule: every time a thunderstorm starts, I go inside.
Quando a trovoada começar, eu vou para dentro de casa.
Here começar is the future subjunctive. This usually refers to a specific future situation: when that storm starts (on that future occasion), I will go inside.

In many informal contexts, some speakers still use the present where, strictly speaking, grammar books would prefer the future subjunctive, but the difference above is a good guideline.

Does eu vou here mean I go or I will go?

In this particular sentence, the most natural reading is habitual:

Quando a trovoada começa, eu vou para dentro de casa.
→ When the thunderstorm starts, I go inside the house. (every time it happens)

Context could also make it sound more like a future response, and then English might prefer I’ll go inside, but in Portuguese the form vou itself is present indicative, not a future tense.

If you wanted a clearer one‑time future idea, you would more likely say something like:

Quando a trovoada começar, vou entrar dentro de casa.
Quando a trovoada começar, irei para dentro de casa.

Can I change the word order to Eu vou para dentro de casa quando a trovoada começa, and what happens to the comma?

Yes, you can change the order. Both are correct:

Quando a trovoada começa, eu vou para dentro de casa.
Eu vou para dentro de casa quando a trovoada começa.

Regarding the comma:

When the quando‑clause comes first, a comma before the main clause is standard:
Quando a trovoada começa, eu vou para dentro de casa.
When the main clause comes first, the comma is usually omitted in European Portuguese:
Eu vou para dentro de casa quando a trovoada começa.

So the word order is flexible, but the comma placement changes with it.