W domu nie ma nikogo.

W domu means at home or in the house/home.

• w = in / at
• domu = a form of dom (house / home)

In this sentence, English usually translates it naturally as at home:
W domu nie ma nikogo. = There is nobody at home.

Because after w when talking about a location, Polish uses the locative case.

So:

• dom = base form, house/home
• w domu = in/at home

This is a very common pattern:

• w sklepie = in the shop
• w pracy = at work
• w domu = at home

Here nie ma means there isn’t / there are not.

In Polish, when talking about whether something or someone is present somewhere, the negative form is very often nie ma:

• Jest ktoś w domu. = Someone is at home.
• Nie ma nikogo w domu. = There is nobody at home.

So in this sentence, nie ma expresses absence.

Because nikogo is the form required here.

• nikt = no one / nobody in the basic form
• nikogo = a different case form, used after nie ma

Polish does not say W domu nie ma nikt.
It says W domu nie ma nikogo.

This is something learners see a lot with nie ma:

• Nie ma chleba. = There is no bread.
• Nie ma ludzi. = There are no people.
• Nie ma nikogo. = There is nobody.

Yes, it is a double negative by English standards, but in Polish this is normal and required.

Polish negative words usually go together with a negative verb:

• Nikt nie przyszedł. = Nobody came.
• Nic nie wiem. = I know nothing / I don’t know anything.
• Nie ma nikogo. = There is nobody.

So nie ma nikogo is the correct Polish structure.
You should not try to make it match English logic.

In this sentence:

• domu is locative singular
• nikogo is genitive

Why?

• w
  • location usually takes the locative: w domu
• nie ma usually takes the genitive for the thing or person that is absent: nikogo

So the structure is:

• W domu = location
• nie ma nikogo = there is nobody

Yes. Polish word order is more flexible than English.

All of these are possible:

• W domu nie ma nikogo.
• Nikogo nie ma w domu.
• Nie ma nikogo w domu.
• W domu nikogo nie ma.

They all mean roughly the same thing, but the emphasis changes slightly.

For example:

• W domu nie ma nikogo focuses first on at home
• Nikogo nie ma w domu puts extra emphasis on nobody

The version you were given is very natural and neutral.

It can mean either, depending on context.

In many everyday situations, w domu is best translated as at home. That is the most natural English version here:

• W domu nie ma nikogo. = There is nobody at home.

But literally it is also in the house/home.

So the exact English wording depends on context and style, but the Polish is the same.

No. That is not correct Polish.

Polish does not use jest nikt for there is nobody.
Instead, it uses the negative structure:

• W domu nie ma nikogo.

A useful contrast is:

• W domu jest ktoś. = Someone is at home.
• W domu nie ma nikogo. = No one is at home.

So the positive and negative patterns are not built in exactly the same way.

A natural positive version would be:

• W domu jest ktoś. = Someone is at home.

You could also say:

• Ktoś jest w domu. = Someone is at home.

But if you want to keep the structure close to the original sentence, W domu jest ktoś is the clearest comparison with W domu nie ma nikogo.

In this expression, nie ma works as a fixed existential phrase meaning there is not / there are not.

So learners should think of nie ma as one unit used to talk about absence:

• Na stole nie ma książki. = There is no book on the table.
• W szkole nie ma nauczyciela. = The teacher is not at school.
• W domu nie ma nikogo. = There is nobody at home.

The important thing is not to translate it word by word, but to recognize the whole pattern.

No. Nikogo is a general case form of nikt, so it appears in other constructions too.

For example:

• Nie widzę nikogo. = I don’t see anyone / I see nobody.
• Nie znam nikogo tutaj. = I don’t know anyone here.

But nie ma nikogo is one of the most common places learners meet it, so it is worth memorizing as a whole phrase.