Для аренды машины нужны паспорт и права.

Russian often expresses need with the pattern needed are X and Y, rather than I need / you need.

So this sentence is literally close to:

For the rental of a car, needed are a passport and a license.

The things that are needed — паспорт и права — are the grammatical subject of the sentence.

A more English-like way to think about it is:

• To rent a car, a passport and license are required.

Russian also can say:

• Вам нужны паспорт и права. = You need a passport and a license.

But in your sentence, the speaker is making a general statement, so there is no explicit you.

Because нужны agrees with паспорт и права, which together form a plural subject.

Compare:

• Нужен паспорт. = A passport is needed.
• Нужны права. = A license / documents are needed.
• Нужны паспорт и права. = A passport and license are needed.

So:

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

Since there are two things, Russian uses the plural: нужны.

They are in the nominative case.

That is because they are the things that are needed — the grammatical subject of the sentence.

You can test this by comparing:

• Паспорт нужен.
• Права нужны.

In your sentence, the two subjects are simply joined together:

• паспорт и права

Both are nominative.

The preposition для requires the genitive case.

So:

• аренда → аренды

That is why you get:

• для аренды = for renting / for the rental

This is a very common pattern:

• для работы = for work
• для поездки = for a trip
• для аренды = for rental / for renting

Because машины here is also part of a genitive phrase.

The structure is:

• для аренды машины

Literally: for the rental of a car

So:

• аренды is genitive after для
• машины is genitive because it depends on аренды: rental of a car

If you said арендовать машину, then you would use the accusative:

• арендовать машину = to rent a car

But here the sentence uses the noun аренда rather than the verb арендовать, so the form changes.

They are very close in meaning, but the style is a little different.

• Для аренды машины нужны паспорт и права. = To rent a car, a passport and license are required. This sounds compact and informational.

• Чтобы арендовать машину, нужны паспорт и права. = In order to rent a car, a passport and license are needed. This sounds a bit more verbal and explicit.

Both are natural. The version with для + noun is common in notices, rules, and formal information.

In Russian, права in this meaning is normally a plural form.

Literally, права can mean rights, but in everyday usage it also means driver’s license or driving license documents.

Very often people say:

• водительские права = driver’s license

But in context, права alone is enough.

So even though English uses a singular word, Russian uses a plural noun here. That is why you say:

• Права нужны not
• Право нужно

Yes, absolutely.

That version is also correct:

• Паспорт и права нужны для аренды машины.

The meaning is the same. The difference is mostly in emphasis:

• Для аренды машины нужны паспорт и права. Focuses first on the situation: for renting a car...

• Паспорт и права нужны для аренды машины. Focuses first on the required documents: a passport and license are needed...

Russian word order is more flexible than English word order, especially when the cases already show the grammatical roles.

Not exactly omitted, but it could be added if you wanted to make the sentence more personal.

The current sentence is a general rule:

• Для аренды машины нужны паспорт и права.

If you want to say you need, you could say:

• Для аренды машины вам нужны паспорт и права.

That means:

• To rent a car, you need a passport and a license.

Without вам, the sentence sounds more neutral and general, like a policy statement.

Because Russian has no articles.

So паспорт can mean:

• a passport
• the passport

and машины can mean:

• of a car
• of the car

The exact meaning depends on context.

In this sentence, English naturally uses a passport and a driver’s license because it is talking about required documents in general.

Both are possible, but they have slightly different styles.

• машина = car, everyday and common
• автомобиль = automobile/car, more formal

So:

• Для аренды машины... sounds natural and conversational.
• Для аренды автомобиля... sounds a bit more formal or official.

In many real-life situations, either one could appear.

Yes, but the structure would be different.

With нужны, the sentence agrees with the things needed:

• Нужны паспорт и права.

With нужно, Russian often uses an impersonal construction, for example:

• Для аренды машины нужно иметь паспорт и права. = To rent a car, it is necessary to have a passport and license.

So:

• нужны паспорт и права = a passport and license are needed
• нужно иметь паспорт и права = it is necessary to have a passport and license

Both are correct, but they are built differently.

A very useful pattern is:

Для + genitive + нужны / нужен / нужна / нужно + nominative

In your sentence:

• Для аренды машины = for renting a car
  
  • both nouns are in the genitive phrase
• нужны
  • agrees with the thing(s) needed
• паспорт и права
  • nominative, because these are the things required

You can make many similar sentences:

• Для поездки нужен паспорт.
• Для работы нужны документы.
• Для входа нужен билет.

This is a very common way to say that something is required for a particular purpose.