Learn Hardcore Portuguese (Brazil): Ela esqueceu o celular no banheiro e voltou rápido para pegar. - She forgot her cellphone in the bathroom and went back quickly to get it.

Portuguese often uses the definite article (o, a, os, as) instead of a possessive when the owner is obvious from context.
Since the subject is ela and we’re talking about celular, listeners automatically understand it’s her phone.
Ela esqueceu o celular is more natural than Ela esqueceu seu celular, which is grammatically correct but sounds more formal or written, and seu can even be ambiguous (“her” or “his” or “your”).

What’s the difference between esquecer and esquecer-se de? Why is there no de here?

There are two common patterns:

esquecer + direct object
- Ela esqueceu o celular. = She forgot her phone.
  Here esquecer is a normal transitive verb, no de.
esquecer-se de + thing / action
- Ela se esqueceu do celular.
- Ela se esqueceu de pegar o celular.

In Brazil, esquecer + object (no se, no de) is very common and perfectly correct. The reflexive form with se and de sounds a bit more formal or careful.

How does no banheiro work grammatically? What does no mean?

No is a contraction of em + o = no (in/on/at + the, masculine singular).
So no banheiro literally is em o banheiro → no banheiro = “in the bathroom.”
If the noun were feminine, it would be em + a = na, e.g. na mesa (“on the table”).

Why is it voltou rápido and not voltou rapidamente?

In Brazilian Portuguese, adjectives like rápido, devagar, lento, etc. are very commonly used as adverbs, especially in speech.
So voltou rápido is the natural, everyday way to say “she came back quickly.”
voltou rapidamente is correct but sounds more formal, careful, or written.

Is rápido agreeing with anything here? Should it be rápida because of ela?

No. In voltou rápido, rápido is functioning as an adverb modifying the verb voltou, so it does not change for gender.
If it were an adjective describing a noun, it would agree:

Ela deu uma volta rápida. (“She took a quick walk/loop.”) – here rápida matches volta (feminine).

Why is there no word after pegar? Shouldn’t it be para pegar o celular?

The direct object (o celular) is omitted because it’s obvious from context; Portuguese often drops repeated words like this.
Fully explicit would be: Ela esqueceu o celular no banheiro e voltou rápido para pegar o celular.
That sounds repetitive, so speakers normally just say para pegar, and listeners understand it means “to get it (the phone).”

Would voltou para pegar mean the same as voltou e pegou?

Not exactly.

voltou para pegar focuses on the purpose of coming back: “she came back to get it.” It doesn’t explicitly say whether she actually got it.
voltou e pegou states two completed actions: “she came back and (actually) got it.”

In everyday conversation, context usually implies that she did get it, but grammatically the first only states the intention.

Can I say voltou rápido para pegá-lo? Where does the pronoun go in Brazilian Portuguese?

Yes: Ela esqueceu o celular no banheiro e voltou rápido para pegá-lo. is correct and a bit more formal.
With infinitives, Brazilian Portuguese usually attaches the pronoun with a hyphen: pegar + o → pegá-lo (note the accent shift).
In Brazilian Portuguese, using pronouns this way is common in writing; in casual speech, people more often omit the object (para pegar) or say para pegar o celular instead of pegá-lo.

Could I change the word order, like Ela esqueceu o celular e voltou rápido no banheiro para pegar?

You could say Ela esqueceu o celular e voltou rápido no banheiro para pegar, and people would understand, but it’s less natural.
Normally, the location phrase stays right after the verb that it belongs to:

Ela esqueceu o celular no banheiro (she forgot it in the bathroom).
Ela voltou rápido para pegar.

If you put no banheiro after voltou, it can sound like she “came back quickly in the bathroom,” which is a bit odd or ambiguous. The original sentence is the clearest.

Could I use another verb instead of pegar, like buscar or apanhar?

Yes, but the nuance changes and some options are more regional:

pegar – very common, neutral: to get, pick up, grab.
- voltou rápido para pegar (o celular).
buscar – often implies going somewhere to get something/someone. Still fine here:
- voltou rápido para buscar o celular.
apanhar – “to pick up” in European Portuguese; in Brazil it mainly means “to get hit / be beaten,” so Brazilians don’t usually use apanhar for “pick up (an object).”

In Brazilian Portuguese, pegar or buscar are the natural choices.

Why is the simple past (esqueceu, voltou) used instead of something like tinha esquecido?

In Brazilian Portuguese, the simple past (pretérito perfeito) is the normal tense for single, completed actions in the past, just like here: she forgot once, then came back once.
Ela tinha esquecido o celular... (“She had forgotten the phone…”) would be used if you were setting up a background situation relative to another past event, e.g.:

Ela tinha esquecido o celular no banheiro quando o garçom chamou.
(“She had forgotten her phone in the bathroom when the waiter called.”)