Библиотека находится далеко от моего дома.

Not exactly. Russian usually does not use a present-tense word for is / am / are in simple sentences.

So instead of saying something like The library is far from my house with a verb to be, Russian often either:

• leaves the verb out completely, or
• uses a more specific verb such as находиться, which means to be located or to be situated.

So Библиотека находится далеко от моего дома is literally something like:

The library is located far from my house.

This sounds a bit more formal or explicit than simply leaving the verb out.

Yes, absolutely.

Библиотека далеко от моего дома is very natural and common. It means the same basic thing: The library is far from my house.

The version with находится:

• sounds a little more formal,
• emphasizes location,
• is common in neutral or written-style Russian.

So both are correct:

• Библиотека далеко от моего дома.
• Библиотека находится далеко от моего дома.

Находится is the 3rd person singular present tense of находиться.

In this sentence it means:

• is located
• is situated
• sometimes just is

So:

• библиотека находится = the library is located

It comes from a verb that historically relates to finding oneself / being situated somewhere, but in modern usage here it is best understood simply as to be located.

Because библиотека is the subject of the sentence.

It is the thing being described:

• Библиотека = the library
• what is being said about it?
  It is located far from my house.

Subjects in Russian normally appear in the nominative case, which is why you see библиотека.

Because the preposition от requires the genitive case.

So after от:

• мой becomes моего
• дом becomes дома

That is why you get:

• от моего дома = from my house / away from my house

This is a very common Russian pattern:

• от брата = from the brother
• от школы = from the school
• от моего дома = from my house

Because in Russian, distance is often expressed with далеко от + genitive.

So even though English learners may think of from mainly as movement, Russian uses от very naturally to mark a point of reference for distance:

• далеко от дома = far from the house
• недалеко от метро = not far from the metro
• близко от берега = close to / near the shore
  though близко к is also common in some contexts

So here от does not mean that the library is moving away. It just marks the place used as the reference point.

Because далеко is an adverb, and here Russian needs an adverb, not an adjective.

The sentence is describing where the library is located, not describing the library itself as a noun with an adjective.

Compare:

• далёкая библиотека = a distant/faraway library
  Here далёкая is an adjective describing library.

But in your sentence:

• библиотека находится далеко = the library is located far away
  Here далеко describes the location relation, so it is an adverb.

This is very similar to English:

• a distant library = adjective
• the library is far away = adverbial meaning

Because мой has to agree with дом, and both are in the genitive singular after от.

Here is the pattern:

• nominative: мой дом = my house
• genitive: моего дома = of/from my house

Since дом is:

• masculine
• singular
• genitive

the possessive word must match it:

• моего

So от моего дома is a matched genitive phrase.

Because дом is in the genitive singular after от.

Its basic form is:

• дом = house, home

But after от it changes:

• от дома = from the house / from home

With the possessive:

• от моего дома = from my house

This is just normal case inflection. Many masculine nouns add -а in the genitive singular.

Russian word order is fairly flexible, but not completely random.

The neutral order here is:

• Библиотека находится далеко от моего дома.

This is the most straightforward way to say it.

You could also rearrange the sentence for emphasis, for example:

• Далеко от моего дома находится библиотека.

This sounds more literary or emphatic, as if highlighting the distance first.

Or:

• Библиотека далеко от моего дома.

Also very natural, especially in speech.

So yes, the word order can change, but the original version is a good neutral model for learners.

Yes, that is a very good idea.

далеко от + genitive is a very common pattern meaning far from.

Examples:

• далеко от центра = far from the center
• далеко от работы = far from work
• далеко от меня = far from me

Learning it as a chunk helps because you automatically remember two things:

1. далеко is the normal word for far
2. от is followed by the genitive

Not if you want the meaning far from my house.

• далеко от моего дома = far from my house
• у моего дома = by my house / near my house

So у changes the meaning completely.

Compare:

• Библиотека находится далеко от моего дома.
  The library is far from my house.

• Библиотека находится у моего дома.
  The library is by my house.

So for distance meaning far from, use далеко от.

A few small things may interest learners:

• Библиотека → stress on the last е: библиоте́ка
• находится → stress on хо́: нахо́дится
• далеко → stress on the last syllable: далеко́
• моего is often pronounced closer to маево / миво in fast speech than the spelling might suggest, though learners should still spell it моего

A rough stress guide:

• библиоте́ка нахо́дится далеко́ от моего́ дома

Stress matters in Russian, so it is worth learning words together with stress when possible.

Because библиотека is singular, the verb also stays singular:

• Библиотека находится... = The library is located...

If the subject were plural:

• Библиотеки находятся далеко от моего дома.
  = The libraries are far from my house.

So the verb changes from:

• находится = singular to
• находятся = plural

This is normal subject-verb agreement in Russian.

Yes, but in everyday speech many speakers would probably prefer the shorter version:

• Библиотека далеко от моего дома.

The version with находится is still correct and natural, but it can sound a bit more formal, descriptive, or textbook-like.

So as a learner, it is useful to know both:

• shorter everyday style: Библиотека далеко от моего дома.
• more explicit/location-focused style: Библиотека находится далеко от моего дома.

Both are good Russian.