W kuchni jest dużo dymu, więc otwórz okno.

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

• kuchnia = kitchen
• w kuchni = in the kitchen

So:

• kuchnia is the basic dictionary form
• kuchni is the locative singular form

This is very common:

• w domu = in the house
• w szkole = at school
• w pracy = at work

Jest is the 3rd person singular form of być (to be). In this sentence it works like there is in English:

• W kuchni jest dużo dymu = There is a lot of smoke in the kitchen

Polish often uses jest in sentences that introduce the existence or presence of something somewhere.

Compare:

• Na stole jest książka = There is a book on the table
• W pokoju jest zimno = It is cold in the room

Because dużo normally requires the genitive case.

• dym = smoke
• dymu = genitive form of dym

So:

• dużo dymu = a lot of smoke
• mało czasu = little time
• sporo ludzi = quite a lot of people

This is a very important pattern in Polish: quantity words like dużo, mało, trochę, sporo are usually followed by the genitive.

In Polish, expressions like dużo dymu are usually treated grammatically as a kind of quantity phrase, and the verb often stays in the singular neuter-style form like jest.

So Polish says:

• jest dużo dymu
• było dużo ludzi
• będzie dużo pracy

Even though English may feel as if the idea is plural or mass-like, Polish commonly uses singular jest/było/będzie in this structure.

Yes. That would also be correct.

The difference is:

• W kuchni jest dym = There is smoke in the kitchen
• W kuchni jest dużo dymu = There is a lot of smoke in the kitchen

So dużo dymu adds emphasis to the amount.

Więc means so, therefore, or thus.

In this sentence:

• W kuchni jest dużo dymu, więc otwórz okno.
• There is a lot of smoke in the kitchen, so open the window.

It connects the cause and the result:

• cause: there is a lot of smoke
• result/action: open the window

A closely related word is więc's more formal cousin zatem, but więc is very common in everyday speech.

Because in Polish, więc is normally preceded by a comma when it connects two clauses.

So this punctuation is standard:

• W kuchni jest dużo dymu, więc otwórz okno.

This is similar to English, where a comma often appears before so when it links two independent parts.

Because otwórz is the imperative form: open!

• otworzyć = to open
• otwórz = open! (said to one person, informally)

So the sentence gives a command or instruction:

• otwórz okno = open the window

This is the normal 2nd person singular imperative form.

It comes from otworzyć, which is the perfective verb meaning to open.

The forms are:

• otworzyć = to open
• otworzę = I will open / I open
• otwórz = open! (imperative)

The stem changes a bit, which is normal in Polish verbs.

This is about aspect, which is very important in Polish.

• otworzyć = perfective
• otwierać = imperfective

Their imperatives are:

• otwórz = open it / open it now / do the complete action
• otwieraj = be opening / open repeatedly / start opening in an ongoing sense

In this sentence, the speaker wants one complete action with a clear result: the window should end up open. That is why otwórz is the natural choice.

So:

• otwórz okno = open the window
• otwieraj okno might sound more like go ahead and open it, keep opening it, or something less neutral in context

Because okno is the accusative singular, and for many neuter nouns the accusative singular is the same as the nominative singular.

• okno = window
• otwórz okno = open the window

Here okno is the direct object of the verb otwórz.

Compare with another neuter noun:

• drzwi are plural-only, so they behave differently
• but with a normal neuter noun:
  • mam jabłko = I have an apple
  • zjedz jabłko = eat the apple

It is not stated explicitly, because Polish often leaves the subject out when it is clear from the verb form.

• otwórz already means you open! addressed to one person informally

So Polish does not need a separate word for you here.

If you added ty, it would sound more emphatic:

• Ty otwórz okno = You open the window

That would usually imply contrast or emphasis.

Yes. Otwórz is the singular informal command, used to one person you address as ty.

If you wanted other versions:

• otwórz okno = open the window (to one person, informal)
• otwórzcie okno = open the window (to more than one person, informal)
• proszę otworzyć okno = please open the window (polite/formal)
• niech pan/pani otworzy okno = please open the window (formal, literally let sir/madam open the window)

Not completely. Polish word order is flexible, but the given sentence is very natural.

• W kuchni jest dużo dymu, więc otwórz okno.

This order sounds normal because it presents:

1. the situation
2. the consequence

You could rearrange parts for emphasis, but not all versions sound equally natural. For example:

• Otwórz okno, bo w kuchni jest dużo dymu. = Open the window, because there is a lot of smoke in the kitchen.

This is also very natural, but it uses bo (because) instead of więc (so).

They express opposite directions of logic:

• bo = because
• więc = so / therefore

Compare:

• W kuchni jest dużo dymu, więc otwórz okno. = There is a lot of smoke in the kitchen, so open the window.

• Otwórz okno, bo w kuchni jest dużo dymu. = Open the window, because there is a lot of smoke in the kitchen.

Both are natural. The difference is just which clause comes first and which connector you use.

Usually dym means smoke. It is the normal word for smoke from fire, burning food, etc.

If you specifically mean chemical fumes or vapors, Polish may use other words depending on context, such as:

• opary = fumes, vapors
• dym = smoke

So in a kitchen context, dym is exactly the expected word.

Yes, absolutely. It sounds natural and idiomatic.

• W kuchni jest dużo dymu is a normal way to describe the situation.
• więc otwórz okno is a normal everyday command.

A speaker might also say:

• W kuchni jest pełno dymu, więc otwórz okno.
• Otwórz okno, bo w kuchni jest dużo dymu.

But your original sentence is fully natural Polish.