Ho bisogno di una risposta immediata.

Yes. Ho is the 1st person singular of avere (to have), so it literally means I have.

In Italian, the idea of to need is often expressed with the structure:

avere bisogno di = to need
Literally: to have need of

So:

• Ho bisogno di una risposta immediata
  = literally I have need of an immediate answer = natural English I need an immediate answer

This is a very common and normal way to say I need in Italian.

Bisogno is a noun, not a verb. It means need.

So in this sentence:

• Ho = I have
• bisogno = need

Together, ho bisogno means I need or more literally I have need.

You will often see it in expressions like:

• Ho bisogno di aiuto = I need help
• Abbiamo bisogno di tempo = We need time
• Hai bisogno di qualcosa? = Do you need something?

Because the fixed expression is:

avere bisogno di + noun / infinitive / pronoun

So di is required after bisogno when you say what is needed.

Examples:

• Ho bisogno di acqua = I need water
• Ho bisogno di una pausa = I need a break
• Ho bisogno di dormire = I need to sleep

In your sentence:

• di una risposta immediata = of an immediate answer

So the full structure is:

Ho bisogno di + una risposta immediata

Una is the feminine singular indefinite article, meaning a or an.

• una risposta = an answer

You usually include the article in Italian when you mean a/an specific countable thing.

So:

• Ho bisogno di una risposta immediata = I need an immediate answer

If you said just Ho bisogno di risposta, that would sound unnatural in standard Italian in this context.

Because risposta is a feminine singular noun.

In Italian, articles and adjectives must agree with the noun in gender and number.

Since risposta is feminine singular, it takes:

• una = feminine singular a/an
• immediata = feminine singular form of immediato

So:

• una risposta immediata

A good clue is the ending -a, which often marks a feminine singular noun, though not always.

In Italian, many adjectives come after the noun, and that is the most normal position here.

So:

• una risposta immediata = an immediate answer

This is the standard and natural order.

Could an adjective sometimes come before the noun in Italian? Yes, but adjective placement can change nuance or sound more literary, emphatic, or stylistic. For this sentence, una risposta immediata is the normal choice.

Because adjectives in Italian must agree with the noun they describe.

The noun is:

• risposta = feminine singular

So the adjective must also be feminine singular:

• masculine singular: immediato
• feminine singular: immediata
• masculine plural: immediati
• feminine plural: immediate

Examples:

• un intervento immediato = an immediate intervention
• una risposta immediata = an immediate answer
• risposte immediate = immediate answers

Yes. Mi serve una risposta immediata is also a very natural sentence.

It literally means something like:

• An immediate answer is necessary for me
• more naturally: I need an immediate answer

The difference is mostly one of structure and nuance:

• Ho bisogno di una risposta immediata = I need an immediate answer
• Mi serve una risposta immediata = I need an immediate answer / An immediate answer is what I need

Both are common.
Ho bisogno di... is a very useful general pattern for learners.

Yes, and in fact you normally do.

Italian often drops subject pronouns when they are clear from the verb form.

So:

• (Io) ho bisogno di una risposta immediata

Both are correct, but Ho bisogno di una risposta immediata is more natural unless you want emphasis, contrast, or clarity.

For example:

• Io ho bisogno di una risposta immediata, non lui.
  = I need an immediate answer, not him.

It is neutral. It works in both formal and informal situations.

It is not rude by itself, but because it is direct, the tone depends on context. If you want to sound more polite, you might add something like:

• Avrei bisogno di una risposta immediata.
  = I would need / I would like an immediate answer
• Avrei bisogno di una risposta il prima possibile.
  = I would need an answer as soon as possible

So your original sentence is perfectly correct, but it can sound firm.

Here is the sentence piece by piece:

• Ho = I have
  1st person singular of avere
• bisogno = need
  a masculine singular noun
• di = of
• una = a/an
  feminine singular indefinite article
• risposta = answer
  feminine singular noun
• immediata = immediate
  feminine singular adjective agreeing with risposta

So the structure is:

Ho bisogno di + noun phrase

And the noun phrase is:

una risposta immediata

You keep bisogno di and change the verb avere.

• Ho bisogno di una risposta immediata = I need an immediate answer
• Hai bisogno di una risposta immediata = You need an immediate answer
• Ha bisogno di una risposta immediata = He/she needs an immediate answer
• Abbiamo bisogno di una risposta immediata = We need an immediate answer
• Avete bisogno di una risposta immediata = You all need an immediate answer
• Hanno bisogno di una risposta immediata = They need an immediate answer

This is a very useful pattern to memorize: avere bisogno di + something

Normally, no. In this expression, you usually use bisogno in the singular:

• Ho bisogno di...

That is the standard pattern.

You may sometimes see i bisogni in other contexts, but then it means needs in the general sense, not the fixed expression avere bisogno di.

For example:

• i bisogni dei bambini = children’s needs

But for I need..., you say:

• Ho bisogno di...