Мне нужен точный ответ на этот вопрос.

Because Russian often expresses need with a dative construction:

• мне = to me / for me
• нужен = is needed

So Мне нужен точный ответ literally means something like To me, an exact answer is needed.

This is a very common Russian pattern:

• Мне нужен совет. = I need advice.
• Ей нужна помощь. = She needs help.
• Нам нужно время. = We need time.

So even though English uses I need..., Russian often uses to me is needed... instead.

Нужен is the short-form adjective of нужный, meaning needed / necessary.

In this sentence, it works as the predicate:

• ответ = answer
• нужен = needed

So the grammar is not based on a normal verb like need in English. Instead, Russian is saying that the answer is needed.

The full idea is:

• Мне нужен ответ. = I need an answer.
• literally: An answer is needed for me.

Because нужен agrees with ответ, which is masculine singular.

Forms of нужный in the short predicate form include:

• нужен — masculine singular
• нужна — feminine singular
• нужно — neuter singular
• нужны — plural

Since ответ is masculine, you say:

• Мне нужен ответ.

Compare:

• Мне нужна книга. = I need a book.
• Мне нужно время. = I need time.
• Мне нужны документы. = I need documents.

Because ответ is the grammatical subject of the sentence.

In this construction:

• мне is in the dative
• the thing needed is in the nominative
• нужен agrees with that nominative noun

So:

• Мне нужен ответ.
• literally: An answer is needed to me / for me.

That is why Russian does not use an accusative object here the way English does in I need an answer.

Because точный is an adjective modifying ответ, and it must agree with it:

• ответ is masculine singular nominative
• so the adjective is also masculine singular nominative: точный

Compare:

• точный ответ — exact answer
• точная информация — exact information
• точное число — exact number
• точные данные — exact data

They are close, but not identical.

• точный = exact, precise
• правильный = correct, right

So:

• точный ответ emphasizes precision and exactness
• правильный ответ emphasizes correctness

Sometimes both could fit, but they are not always interchangeable. In your sentence, точный ответ suggests the speaker wants a precise, definite answer.

Because Russian uses the fixed expression ответ на вопрос = answer to a question.

So after ответ, Russian normally uses:

• на + accusative

Examples:

• ответ на письмо = a reply to a letter
• ответ на вопрос = an answer to a question
• ответ на просьбу = a response to a request

Even though English says answer to a question, Russian uses на here.

Because after на in this expression, вопрос is in the accusative case:

• на этот вопрос

But вопрос is masculine and inanimate, and in Russian the accusative of masculine inanimate nouns looks the same as the nominative:

• nominative: этот вопрос
• accusative: этот вопрос

So the form does not change here.

If it were animate, you would see a different pattern. But вопрос is inanimate, so accusative = nominative.

Yes, in a way.

Russian usually omits the present-tense form of to be. English says:

• An answer is needed

Russian simply says:

• Ответ нужен

So in Мне нужен точный ответ на этот вопрос, there is no separate word for is, but the meaning is still there.

This is normal in Russian present-tense sentences.

Yes. Russian word order is flexible, and changing it often changes emphasis rather than the basic meaning.

Your sentence:

• Мне нужен точный ответ на этот вопрос.

This is a natural neutral order.

Other possible orders:

• Точный ответ на этот вопрос мне нужен.
  Emphasizes точный ответ
• На этот вопрос мне нужен точный ответ.
  Emphasizes на этот вопрос

The original version sounds very natural if you are simply stating what you need.

Yes. That would simply mean:

• I need an answer to this question.

Adding точный makes it stronger and more specific:

• Мне нужен ответ... = I need an answer...
• Мне нужен точный ответ... = I need an exact / precise answer...

So точный adds emphasis on the quality of the answer.

The form of нужен would change to agree with the noun.

Examples:

• Мне нужна точная информация. = I need exact information.
  (информация is feminine)

• Мне нужно точное объяснение. = I need an exact explanation.
  (объяснение is neuter)

• Мне нужны точные данные. = I need exact data.
  (данные is plural)

This is the same pattern as in your sentence; only the gender/number changes.