Vull fer una pregunta a la professora.

In Catalan, subject pronouns are often omitted because the verb ending already tells you who the subject is.

• vull = I want
• so jo is not necessary

You can say Jo vull fer una pregunta a la professora, but that sounds more emphatic, like I want to ask a question.

Vull is the 1st person singular present tense of voler, which means to want.

So:

• vull = I want
• voler = to want

This is an irregular verb, so the form vull does not look very similar to the infinitive voler.

Catalan often uses fer una pregunta, literally to make a question / ask a question, just like English can say ask a question.

But Catalan can also use preguntar:

• Vull fer una pregunta a la professora.
• Vull preguntar una cosa a la professora.

Both are natural.
Fer una pregunta focuses on the noun question, while preguntar uses the verb to ask more directly.

Yes, but it is slightly incomplete unless you add what you want to ask.

For example:

• Vull preguntar una cosa a la professora. = I want to ask the teacher something.
• Vull preguntar a la professora si hi haurà examen. = I want to ask the teacher whether there will be an exam.

So fer una pregunta works well when you want to mention a question as a noun.

That a marks the indirect object, the person the question is directed to.

So in:

• fer una pregunta a la professora

the teacher is the person to whom the question is being asked.

This is very common in Catalan:

• Donaré el llibre a la Maria. = I’ll give the book to Maria.
• Vull dir una cosa al professor. = I want to say something to the teacher.

Because a + la stays a la in Catalan.

Only some combinations contract:

• a + el = al
• de + el = del

So:

• a la professora = correct
• al professor = correct

But there is no contraction with la here.

Catalan often uses the definite article with professions or roles when referring to a specific person.

So:

• la professora = the teacher
• el professor = the teacher (male)

In this sentence, it means a specific teacher, so the article is natural and expected.

Yes. Catalan nouns often show gender clearly.

• professor = male teacher
• professora = female teacher

The article also matches:

• el professor
• la professora

So the sentence tells you the teacher is female.

Because pregunta is a countable noun, and here it means a question, not the question.

• una pregunta = a question
• la pregunta = the question

So fer una pregunta is the natural way to say ask a question.

The normal word order here is very natural:

• Vull fer una pregunta a la professora.

Catalan does allow some flexibility, but changing the order usually adds emphasis or sounds more marked.

For example:

• A la professora, li vull fer una pregunta.
  This emphasizes to the teacher.

For learners, the original order is the safest and most neutral.

Yes. Li is the indirect object pronoun meaning to him / to her / to you (formal).

So you could say:

• Li vull fer una pregunta. = I want to ask her a question.

You can also use both together:

• Li vull fer una pregunta a la professora.

That doubling is common in Catalan, especially in spoken language.

In standard Catalan, vull is usually pronounced approximately like booly said very quickly, with a final ll sound.

A rough guide:

• v often sounds like b
• u is like oo
• ll can sound like the lli in some pronunciations of million, though pronunciation varies by dialect

So vull is roughly bool / booy / booly, depending on the variety of Catalan.

It is neutral and polite. Nothing in it is especially formal or especially casual.

• Vull fer una pregunta a la professora. is a normal, correct sentence.

If you were directly addressing the teacher, you might phrase things differently to sound more polite, for example:

• Puc fer una pregunta? = May I ask a question?

But as a statement, the original sentence is perfectly natural.