Learn Hardcore Portuguese (Portugal): Depois de um pesadelo, eu fico acordado a olhar para o teto. - After a nightmare, I stay awake staring at the ceiling.

Do is the contraction of de + o (of/from the), so it is definite: depois do pesadelo = after the nightmare (a specific one we both know about).

De um uses the indefinite article: depois de um pesadelo = after a nightmare (any nightmare, not a particular one).

So in your sentence, the idea is general: whenever I have a nightmare / after a nightmare. If you were talking about a specific nightmare already mentioned, you could say:

Depois do pesadelo de ontem, eu fiquei acordado…
After yesterday’s nightmare, I stayed awake…

In European Portuguese, in everyday speech and informal writing, de um is often contracted to dum:

Depois dum pesadelo, fico acordado…

Can we say depois de before a verb as well? How would I say After having a nightmare?

Yes. Depois de can come before a noun or a verb in the infinitive.

With a noun (your original sentence):

Depois de um pesadelo, eu fico acordado…
After a nightmare, I stay awake…

With a verb:

Depois de ter um pesadelo, eu fico acordado…
After having a nightmare, I stay awake…
Depois de acordar de um pesadelo, fico acordado…
After waking up from a nightmare, I stay awake…

In all these, depois de is followed either by a noun (um pesadelo) or by a verb in the infinitive (ter, acordar).

Why use eu fico acordado instead of just eu acordo?

Acordar = to wake up (the action of going from sleeping to being awake).
Ficar acordado = to stay/remain awake (a state that continues).

So:

Depois de um pesadelo, eu acordo.
After a nightmare, I wake up. (You were asleep and you wake up. It says nothing about whether you go back to sleep.)
Depois de um pesadelo, eu fico acordado.
After a nightmare, I stay awake. (You wake up and then you remain awake; you don’t fall asleep again.)

Your sentence is specifically about not going back to sleep, so ficar acordado is the natural choice.

What is the difference between ficar acordado and estar acordado?

Both end up describing being awake, but the focus is different:

estar acordado – describes the current state:
Estou acordado. = I am awake.
ficar acordado – focuses on becoming and remaining awake:
Fico acordado a noite toda. = I stay awake all night.
(I enter and maintain that state.)

In your sentence:

Depois de um pesadelo, eu fico acordado…
Emphasises that because of the nightmare, you end up staying awake (and not going back to sleep).

If you said:

Depois de um pesadelo, eu estou acordado…

it sounds more like a neutral statement of fact about your state after the nightmare, without the idea of remaining awake so strongly.

Does acordado change for female speakers?

Yes. Acordado is an adjective/participle and it agrees in gender and number with the subject.

Male speaker (singular):
Eu fico acordado.
Female speaker (singular):
Eu fico acordada.
Group of only men or mixed group:
Nós ficamos acordados.
Group of only women:
Nós ficamos acordadas.

So if you’re female, you would say:

Depois de um pesadelo, eu fico acordada a olhar para o teto.

Why is it a olhar instead of olhando?

European Portuguese rarely uses the -ndo gerund (like olhando) in everyday speech. Instead, it prefers a + infinitive to express ongoing actions:

a olhar = looking
a ler = reading
a comer = eating

So:

Fico acordado a olhar para o teto.
= I stay awake looking at the ceiling.

In Brazilian Portuguese, the gerund is much more common:

Fico acordado olhando para o teto.

In Portugal, olhando is not wrong, but it sounds either literary, archaic, or regional. The natural European Portuguese form in this sentence is a olhar.

Why do we say olhar para o teto and not olhar o teto or olhar no teto?

In European Portuguese, the usual pattern is:

olhar para algo/alguém = to look at something/someone

So:

olhar para o teto = to look at the ceiling

Other options:

olhar o teto – possible, but much less common and can sound a bit more formal or descriptive (like noticing the ceiling itself, not just staring upwards).
olhar no teto – unusual; would usually be understood as look inside the ceiling or look at something that is in/on the ceiling, and doesn’t fit the natural pattern for “staring at the ceiling”.

For the idea lie in bed and stare at the ceiling, olhar para o teto is the standard expression.

Is teto the same as tecto? I’ve seen both.

Yes, they are the same word; the spelling changed with the spelling reform.

Before the Portuguese spelling reform: tecto (European Portuguese)
After the reform (now official): teto

Brazil already used teto, and now Portugal also officially uses teto. You will still see tecto in older texts, but in modern writing in Portugal you should prefer teto.

Can I omit eu and just say Depois de um pesadelo, fico acordado a olhar para o teto?

Yes. Portuguese is a pro‑drop language: the subject pronoun is often omitted because the verb ending already shows the person.

Eu fico acordado…
Fico acordado…

Both are correct. The version without eu is probably more natural in everyday speech:

Depois de um pesadelo, fico acordado a olhar para o teto.

Why is there a comma after pesadelo? Is it obligatory?

The comma separates an initial adverbial phrase (time expression) from the main clause:

Depois de um pesadelo, (when?)
eu fico acordado a olhar para o teto. (what happens?)

In European Portuguese, it’s standard and recommended to use the comma there, and it’s what you’ll usually see in writing.

If you removed it:

Depois de um pesadelo eu fico acordado…

it’s still understandable and not a serious mistake in informal contexts, but the version with the comma follows the usual punctuation rule more closely.

Could I say após um pesadelo instead of depois de um pesadelo?

You can, but the tone changes a bit.

Depois de um pesadelo, fico acordado…
Very common, neutral, everyday language.
Após um pesadelo, fico acordado…
Grammatically correct, but após tends to sound more formal, written, or literary.

So you can use após, but depois de is the most natural choice in conversational European Portuguese for this sentence.