Learn Hardcore Portuguese (Brazil): Eu quero perguntar algo ao doutor amanhã. - I want to ask the doctor something tomorrow.

In this sentence, doutor is the indirect object (the person receiving the question), so in Portuguese you normally need a preposition, usually a:

perguntar algo a alguém = to ask something to someone

Therefore:

perguntar algo ao doutor = to ask something to the doctor

You cannot say perguntar algo o doutor (missing the preposition); you need a(o) there, so it becomes ao.

Can I say para o doutor instead of ao doutor? Is there a difference?

Yes, you can say:

Eu quero perguntar algo ao doutor amanhã.
Eu quero perguntar algo para o doutor amanhã.

Both are correct and widely used in Brazil.

Subtle points:

a (→ ao) is a bit more traditional / closer to written standard.
para is very common in spoken Brazilian Portuguese and can sound more natural in everyday speech.

For most learners, ao doutor and para o doutor are interchangeable here. In formal writing, you’ll see a/ao more often.

Why is it doutor and not médico? What’s the difference?

In Brazil:

médico = the profession (doctor, physician)
doutor / doutora = a form of address, like saying “Doctor” when you speak to or about a physician (or another highly educated professional).

Examples:

Este médico é muito bom. – This doctor is very good. (talking about his profession)
Eu quero falar com o doutor. – I want to speak with the doctor. (addressing him as “Doctor”)

So in your sentence:

ao doutor ≈ to the doctor (as a respectful title)

You could also say:

Eu quero perguntar algo ao médico amanhã.

That focuses more on the job (the physician), and is also correct, just a slightly different nuance.

If the doctor is a woman, do I change anything?

Yes, you change doutor to doutora, and the article changes to feminine:

Eu quero perguntar algo à doutora amanhã.

Here you get a different contraction:

a + a = à (with grave accent)

So:

masculine: ao doutor (a + o)
feminine: à doutora (a + a)

Do I really need to say Eu? Can I just say Quero perguntar algo ao doutor amanhã?

You can drop eu, and it’s very natural:

Quero perguntar algo ao doutor amanhã.

In Portuguese, the verb ending -o in quero already tells us the subject is eu:

(Eu) quero – I want

So both are correct:

Eu quero perguntar… (slightly more explicit/emphatic)
Quero perguntar… (very common and natural)

Why do we use quero (present tense) if the action will happen amanhã (tomorrow)?

In Portuguese, it’s normal to use the present tense of querer (to want) to talk about future plans:

Eu quero perguntar algo ao doutor amanhã.
= I want to ask the doctor something tomorrow.

You’re stating your current desire/intention about a future action.

You could also use the future tense:

Perguntarei algo ao doutor amanhã. – I will ask the doctor something tomorrow.

But that sounds more formal / written and is less common in everyday speech. A more natural spoken option is:

Vou perguntar algo ao doutor amanhã. – I’m going to ask the doctor something tomorrow.

What’s the difference between perguntar and fazer uma pergunta?

They are very close in meaning:

perguntar = to ask
fazer uma pergunta = to make/ask a question

Your sentence:

Eu quero perguntar algo ao doutor amanhã.

Could also be expressed as:

Eu quero fazer uma pergunta ao doutor amanhã. – I want to ask the doctor a question tomorrow.

Small nuance:

perguntar algo focuses on the content (to ask something).
fazer uma pergunta focuses on the act (to ask a question).

Both are completely natural here.

What does algo mean exactly, and can I say alguma coisa instead?

Algo means “something”.

You can definitely say:

Eu quero perguntar algo ao doutor amanhã.
Eu quero perguntar alguma coisa ao doutor amanhã.

Differences:

algo is a bit shorter and often sounds slightly more neutral or formal.
alguma coisa is very common in everyday speech, and can feel a bit more casual/colloquial.

Both are fine in Brazilian Portuguese.

Why is amanhã at the end? Can it go in other positions?

Time words like amanhã (tomorrow) are very flexible in Portuguese. All these are possible:

Eu quero perguntar algo ao doutor amanhã.
Amanhã eu quero perguntar algo ao doutor.
Eu quero, amanhã, perguntar algo ao doutor. (more formal/emphatic)

Most natural in spoken language:

Amanhã eu quero perguntar algo ao doutor.
Eu quero perguntar algo ao doutor amanhã.

Putting amanhã at the end (as in your sentence) is perfectly normal and common.

Could I use an indirect object pronoun instead of ao doutor, like lhe?

Yes. You can say:

Eu quero perguntar algo ao doutor amanhã.
Eu quero perguntar algo a ele amanhã. – I want to ask him something tomorrow.
Eu quero lhe perguntar algo amanhã. – I want to ask him/her something tomorrow.

Notes:

lhe is the standard indirect object pronoun for a ele / a ela (to him / to her), but in Brazilian Portuguese, especially spoken, people very often prefer para ele / para ela or pra ele / pra ela instead of lhe.
Using lhe sounds more formal or bookish in much of Brazil.

Is quero perguntar the most natural way to say this, or would Brazilians use something else?

Quero perguntar… is absolutely natural and correct.

Common alternatives Brazilians might also use:

Quero falar com o doutor amanhã. – I want to talk to the doctor tomorrow.
Quero tirar uma dúvida com o doutor amanhã. – I want to clear up a question with the doctor tomorrow.
Quero perguntar uma coisa pro doutor amanhã. – I want to ask the doctor something tomorrow. (very colloquial: pro = para o)

Your original structure is good, clear Portuguese and perfectly usable.

Is quero polite enough when talking about a doctor, or should I say something softer like gostaria de?

Quero is neutral and common, not rude by itself. But in very polite or formal contexts, Brazilians often soften it:

Eu gostaria de perguntar algo ao doutor amanhã.
= I would like to ask the doctor something tomorrow.

So:

Quero perguntar… – straightforward, everyday speech.
Gostaria de perguntar… – more polite / courteous / formal.

Both are correct; choose based on how formal or respectful you want to sound.