Из-под двери идёт холод через маленькую щель.

Из-под is a fixed compound preposition meaning from under.

So из-под двери means from under the door.

It is normally written with a hyphen because it functions as one unit in this meaning. It also takes the genitive case.

Because из-под requires the genitive.

The noun дверь changes like this:

• nominative: дверь
• genitive: двери

So:

• из-под двери = from under the door

A learner may notice that двери can also be nominative plural, but here it is clearly genitive singular because it follows из-под.

In Russian, идти is often used more broadly than English to go.

It can describe things that move/spread/come through in a natural way, such as:

• дым идёт = smoke is coming
• запах идёт = a smell is coming
• тепло идёт = warmth is coming
• холод идёт = cold is coming in / there is cold coming through

So идёт холод does not mean the cold is walking. It means the cold is coming through, entering, or being felt from that direction.

Because холод and холодно are different parts of speech.

• холод = cold, as a noun
• холодно = it is cold, as a predicative word/adverb-like form

In this sentence, холод is the thing that is coming through, so a noun works well:

• идёт холод = cold is coming through

If you said холодно, you would be making a statement about the general condition:

• В комнате холодно = it is cold in the room

Через usually means through, across, or via, depending on context.

Here it means through:

• через маленькую щель = through a small crack/gap

It shows the opening that the cold is passing through.

Because через takes the accusative case.

The phrase is:

• через маленькую щель

Here:

• маленькая becomes маленькую because it is feminine singular accusative
• щель is an inanimate feminine noun of the third declension, and its accusative singular looks the same as its nominative singular

So the case change is visible clearly on the adjective, but not on the noun’s spelling.

No. Russian word order is fairly flexible.

This sentence begins with Из-под двери to set the scene first: that is where the cold is coming from.

A more straightforward English-like order would be:

• Холод идёт из-под двери через маленькую щель.

Both are understandable, but the original version feels natural because Russian often places location/source information early in the sentence.

Slightly, yes, but it is still natural.

The sentence is painting a physical picture:

• the cold is coming from under the door
• specifically through a small gap

So the second phrase adds detail. It is not wrong repetition; it is descriptive.

In everyday speech, a speaker might shorten it if the gap is already obvious.

Yes, very often.

That is another very natural way to express the idea. It means something like:

• There is a draft coming from under the door
• It is blowing from under the door

The difference is roughly this:

• идёт холод emphasizes the cold coming in
• дует emphasizes the draft/blowing air

Both are natural, but дует is especially common when talking about drafts.

Because Russian has no articles.

English must choose between the door and a door, a small crack and the small crack, but Russian does not mark that directly.

So:

• двери can mean the door or a door, depending on context
• маленькую щель can mean a small crack or the small crack

Usually the situation makes it clear which meaning is intended.