Kun jij op mijn vraag antwoorden?

Because this is a yes/no question. In Dutch, yes/no questions usually put the finite verb first.

• Statement: Jij kunt op mijn vraag antwoorden.
• Question: Kun jij op mijn vraag antwoorden?

So the sentence starts with kun because the verb moves to the front to form the question.

With jij/je, Dutch normally drops the -t ending when the verb comes before the subject.

Compare:

• Statement: Jij kunt
• Question: Kun jij?

This happens with many verbs:

• Jij werkt → Werk jij?
• Jij hebt → Heb jij?

So Kun jij ...? is the expected form here.

Both can mean you.

• Kun jij op mijn vraag antwoorden?
• Kun je op mijn vraag antwoorden?

The version with je is usually more neutral and common in everyday speech.
The version with jij sounds a bit more emphatic or contrastive, as if you are stressing you:

• Kun jij op mijn vraag antwoorden, of moet ik iemand anders vragen?

So in many situations, Kun je ...? would sound slightly more natural, but Kun jij ...? is absolutely correct.

Because antwoorden normally goes with the preposition op when you say what someone is answering.

So Dutch says:

• antwoorden op een vraag
• literally something like answer on/to a question

English uses answer a question without a preposition, but Dutch does not work the same way here.

Examples:

• Ik antwoord op je e-mail.
• Hij antwoordde op mijn vraag.

So op mijn vraag antwoorden is the normal pattern.

Because kun is a modal verb (can), and in Dutch, when a modal verb is conjugated, the other verb usually goes to the end in the infinitive.

So:

• Kun = finite/modal verb
• antwoorden = infinitive at the end

This is a very common pattern:

• Ik kan komen.
• Kun jij helpen?
• Wij willen vertrekken.

In your sentence, op mijn vraag comes before the final infinitive, giving:

• Kun jij op mijn vraag antwoorden?

Yes. That is also correct, and very common.

There are two common ways to express this idea:

• op mijn vraag antwoorden
• mijn vraag beantwoorden

The difference is mainly grammatical:

• antwoorden op takes a preposition
• beantwoorden takes a direct object

So:

• Kun jij op mijn vraag antwoorden?
• Kun jij mijn vraag beantwoorden?

Both are natural.
Beantwoorden can sound a little more direct or slightly more formal in some contexts, but both are normal Dutch.

It can be both, depending on context.

Literally, Kun jij ...? asks about ability: Can you ...?
But just like in English, it is also often used as a polite request.

So this sentence can mean:

• Are you able to answer my question?
• Could you answer my question?

In real conversation, it will often function as a request rather than a serious question about ability.

Use u instead of jij/je:

• Kunt u op mijn vraag antwoorden?
• Kunt u mijn vraag beantwoorden?

This is the polite/formal version.

A few levels:

• Kun je op mijn vraag antwoorden? = informal, neutral
• Kun jij op mijn vraag antwoorden? = informal, with extra emphasis on you
• Kunt u op mijn vraag antwoorden? = formal/polite

Dutch usually uses a possessive adjective here, just like English:

• mijn vraag = my question

That is the normal and natural choice.

De vraag van mij is grammatically possible in some special contexts, but it is not the usual way to say my question. In ordinary Dutch, mijn vraag is what you want.