No posto, a enfermeira me chamou pelo meu nome e perguntou se eu tinha dor.

No posto usually means at the clinic/health post (short for posto de saúde) in Brazilian Portuguese.
While posto can also mean a gas station (posto de gasolina) or a job/position, the presence of a enfermeira (the nurse) and questions about pain make the healthcare meaning the natural one here.

Because Portuguese commonly contracts em + o into no:

• em + o = no → No posto = In/At the clinic
  Similarly: em + a = na, em + os = nos, em + as = nas.

No posto is a fronted setting phrase (it tells you where everything happened). The comma is optional but common to separate that introductory detail from the main clause:

• No posto, a enfermeira... = At the clinic, the nurse...

Portuguese normally uses definite articles much more than English does. A enfermeira here means the nurse (a specific nurse in that situation).
You typically don’t drop it in neutral narration. Dropping it can sound unnatural or more like a headline style.

Me is an object pronoun meaning me (the person receiving the action).

• A enfermeira me chamou = The nurse called me
  In Brazilian Portuguese, this placement (me before the verb) is very common in speech and writing.

Yes, chamou-me is grammatically correct, but in Brazil it often sounds more formal/literary. In everyday Brazilian Portuguese, me chamou is generally more natural.

Both are in the pretérito perfeito (simple past), used for completed actions:

• chamar → chamou = called
• perguntar → perguntou = asked
  So the sentence describes two finished actions in the past.

Pelo is the contraction of por + o. In this expression, chamar pelo nome means to call someone by their name.

• pelo meu nome = by my name / using my name
  It implies she addressed you using your name (not just called you over).

Common alternatives include:

• me chamou pelo nome = called me by (my) name (very common)
• me chamou pelo meu nome completo = called me by my full name
• me chamou pelo primeiro nome = called me by my first name
  You might also hear me chamou pelo nome de... in some contexts, but pelo (meu) nome is the standard phrasing.

Yes. Perguntar se... introduces an indirect yes/no question and corresponds closely to English asked if/whether...

• perguntou se eu tinha dor = asked if I had pain

Portuguese often allows dropping subject pronouns, and you could say:

• ...perguntou se tinha dor.
  But including eu makes it extra clear and is very common in Brazilian Portuguese, especially in spoken language.

All are possible, but they express slightly different ideas:

• tinha dor (imperfect) = “was having pain / had pain (as a state)”—common in reported questions like this.
• tive dor (perfect) = “I had pain (at some point / as an event)”—often sounds more like a completed episode.
• estava com dor = “I was in pain” (very common in Brazil and often the most natural everyday way).
  So tinha dor works well as a neutral, reported version of what she asked.

Not necessarily. In Portuguese, dor can be used without an article to mean pain in general:

• tinha dor = had pain
  If you mean a specific pain, you might add something:
• tinha uma dor forte = had a strong pain
• tinha dor no peito = had chest pain