Мне нужна маленькая отвёртка для этого винта.

Because Russian uses a different structure for to need.

Мне нужна маленькая отвёртка literally means something like To me, a small screwdriver is needed.

So:

• я = I nominative
• мне = to me / for me dative

With нужен / нужна / нужно / нужны, the person who needs something is usually in the dative case:

• Мне нужен ключ. = I need a key.
• Тебе нужна помощь. = You need help.

Because нужна agrees with отвёртка, and отвёртка is a feminine singular noun.

Forms of нужен:

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

Examples:

• Мне нужен винт. = I need a screw.
• Мне нужна отвёртка. = I need a screwdriver.
• Мне нужно время. = I need time.
• Мне нужны инструменты. = I need tools.

Because adjectives in Russian must agree with the noun they describe in:

• gender
• number
• case

Here, отвёртка is:

• feminine
• singular
• nominative

So маленький becomes маленькая.

Compare:

• маленький винт — a small screw
• маленькая отвёртка — a small screwdriver
• маленькое окно — a small window
• маленькие детали — small parts

Because it is the thing that is needed, and with this structure the needed thing normally stays in the nominative:

• Мне нужен ключ.
• Мне нужна отвёртка.
• Ему нужны деньги.

So in this sentence:

• мне = dative, the person who needs
• отвёртка = nominative, the thing needed

Because для requires the genitive case.

So:

• этот винт = this screw (nominative)
• для этого винта = for this screw (genitive)

Both words change:

• этот → этого
• винт → винта

This is very common in Russian:

• для дома = for the house
• для машины = for the car
• для этого проекта = for this project

Here для means for.

So для этого винта means for this screw — in other words, the screwdriver is intended to fit or be used with this screw.

It gives the purpose or target:

• ложка для супа = a spoon for soup
• очки для чтения = glasses for reading
• инструмент для ремонта = a tool for repair

Yes — from an English point of view, it feels like something is missing, but Russian often omits the present-tense to be.

So:

• Мне нужна маленькая отвёртка literally does not include a word for is
• but the meaning is still I need a small screwdriver

This is normal in Russian:

• Он дома. = He is at home.
• Она врач. = She is a doctor.

In the present tense, Russian usually leaves out is / am / are.

Yes. Russian word order is more flexible than English, although the basic version here is very natural.

Original:

• Мне нужна маленькая отвёртка для этого винта.

Other possible orders:

• Для этого винта мне нужна маленькая отвёртка.
• Маленькая отвёртка мне нужна для этого винта.

These all mean roughly the same thing, but the emphasis changes:

• starting with для этого винта emphasizes for this screw
• starting with маленькая отвёртка emphasizes the small screwdriver

Because этого is the genitive singular masculine form of этот.

The gender has not changed — it is still masculine. The case changed because of для.

For этот:

• nominative masculine: этот
• genitive masculine/neuter: этого

So:

• этот винт = this screw
• для этого винта = for this screw

The stress is on ё:

• отвёртка

The letter ё always indicates a stressed yo sound. That makes pronunciation easier here.

So it sounds roughly like:

• at-VYORT-ka

In many printed texts, Russian ё is often written as е, so you may also see отвертка, but the pronunciation is still отвёртка.

No. That is incorrect because the adjective and нужен must agree with отвёртка, which is feminine.

Correct:

• Мне нужна маленькая отвёртка.

Incorrect:

• Мне нужен маленький отвёртка.

Both words must match the feminine noun:

• нужна
• маленькая

Not exactly. In everyday speech, learners often see both translated as screw, but Russian can distinguish them.

Very roughly:

• винт often refers to a machine screw/bolt-like fastener
• шуруп often refers to a wood screw or screw that cuts into material

In real life, people are not always perfectly technical, but винт in this sentence is completely natural if the object really is that type of screw.

It sounds completely normal and natural.

Мне нужна маленькая отвёртка для этого винта. is a standard everyday sentence. You could say it:

• at home
• in a workshop
• in a hardware store
• while fixing something

A slightly shorter version is also common if the context is obvious:

• Мне нужна маленькая отвёртка. = I need a small screwdriver.
• Для этого винта нужна маленькая отвёртка. = A small screwdriver is needed for this screw.