Ремонт дома идёт быстро.

It means the repair/renovation of the house or the house renovation.

• ремонт = repair, renovation
• дома = of the house

So this is a noun phrase where the second noun tells you what is being repaired.

Because Russian often uses the genitive case after a noun to show a relationship like of in English.

So:

• дом = house
• дома = of the house

In ремонт дома, the meaning is literally repair of the house.

Not in this sentence.

The form дома can mean different things in Russian, but context tells you which one it is.

Here it means of the house because it depends on ремонт:

• ремонт чего? = repair of what?
• дома = of the house

So in this sentence it is a genitive singular noun form, not at home and not houses.

Yes, literally идти means to go or to walk, but in Russian it is also very commonly used in a broader sense: to go on, to proceed, to be underway, to progress.

So:

• Ремонт дома идёт быстро = The house renovation is going quickly
• more naturally in English: The renovation is progressing quickly

This is a very normal Russian way to talk about a process.

Because Russian normally does not use a present-tense verb meaning is/are the way English does.

English says:

• The renovation is fast
• The renovation is going quickly

Russian often just uses the main verb directly, without a present-tense to be.

Here, идёт already carries the meaning of is going / is in progress.

The subject is ремонт.

You can tell because:

• ремонт is in the nominative form
• идёт is singular
• the sentence is about the repair/renovation

So the structure is basically:

• Ремонт = subject
• дома = dependent noun, of the house
• идёт быстро = is progressing quickly

Because быстро is an adverb, and it modifies the verb идёт.

It tells us how the renovation is going:

• идёт быстро = is going quickly

If you used быстрый, that would be an adjective, and it would describe the noun:

• быстрый ремонт = a quick renovation

So:

• быстрый ремонт = the renovation itself is quick
• ремонт идёт быстро = the renovation is progressing quickly

No, Russian word order is fairly flexible.

Ремонт дома идёт быстро is a neutral, natural order.

You could also hear:

• Ремонт дома быстро идёт
• Быстро идёт ремонт дома

These are grammatically possible, but they may shift the emphasis a little.

The original version is a good neutral model for learners.

It can mean either, depending on context.

ремонт is a broad word that can refer to:

• repair
• renovation
• refurbishment
• maintenance work

So Ремонт дома идёт быстро could mean:

• The repair of the house is progressing quickly
• The house renovation is going quickly

The exact English translation depends on the situation.

Because the sentence describes an ongoing process.

Russian uses the imperfective verb here to show that the renovation is currently in progress.

• идти = imperfective
• идёт = is going, is proceeding

A perfective verb would not work the same way here, because the point is not that the renovation is finished, but that it is underway.

Идёт is pronounced roughly like ee-DYOT.

Important point:

• ё is pronounced like yo
• ё is always stressed

So идёт is not pronounced like idet. It is idyot.

In many printed texts, Russians sometimes write е instead of ё, so you may see идет, but it is still understood and pronounced as идёт here.

A fairly literal translation would be:

• The repair of the house goes quickly

That sounds a bit unnatural in English, so a better natural translation is:

• The house renovation is going quickly
• The renovation is progressing quickly

This is a good example of where Russian and English express the same idea a little differently.

Russian has no articles, so there is no direct equivalent of a or the in the sentence.

Whether you translate it as:

• a house
• the house

depends on context.

In many situations, English would naturally use the:

• The house renovation is progressing quickly

But Russian itself does not mark that distinction with an article.