La biblioteca está bien lejos de mi casa.

Because Spanish uses estar for location.

In this sentence, la biblioteca is being located relative to mi casa, so estar is the normal verb:

• La biblioteca está lejos de mi casa.

Using ser here would sound wrong. Spanish says estar lejos de..., not ser lejos de....

Not in this sentence.

Here, bien is an intensifier, so bien lejos means something like:

• really far
• quite far
• pretty far

So it does not mean well far in the literal sense. This use of bien is common in informal Spanish.

Both are possible, but they do not sound exactly the same.

• muy lejos = neutral, standard, safe in any context
• bien lejos = more informal, more conversational, with extra emphasis

So if you want the most neutral textbook version, use muy lejos:

• La biblioteca está muy lejos de mi casa.

If you want a more colloquial tone, bien lejos works well.

Because lejos normally goes with de when you say what something is far from.

The pattern is:

lejos de + place/person

Examples:

• lejos de Madrid
• lejos del centro
• lejos de mi casa

So de is part of the normal structure.

Because in modern standard Spanish, possessives like mi, tu, su, nuestro usually replace the article.

So you say:

• mi casa
• tu libro
• nuestra ciudad

Not:

• la mi casa
• el tu libro

Those older forms are not standard in modern everyday Spanish.

Because la and mi do the same kind of job: they both introduce the noun.

• la biblioteca = the library
• mi casa = my house/home

Spanish normally uses one determiner before the noun, not two. Since mi is already there, you do not add la.

Here, lejos works like an adverb in the expression estar lejos de.

That is why it does not change:

• La biblioteca está lejos.
• Los museos están lejos.

It stays lejos no matter what the subject is.

Compare that with the adjective lejano / lejana:

• una biblioteca lejana
• unos pueblos lejanos

That is a different structure.

No, that would not sound natural.

For this idea, Spanish normally uses:

• La biblioteca está lejos de mi casa.

Spanish does have lejano / lejana, but it is usually used differently, for example:

• un barrio lejano
• una fecha lejana

So for physical distance from another place, estar lejos de is the expression to remember.

Because bien is modifying lejos.

It is similar to how English says:

• really far
• quite far

So the normal order is:

• bien lejos

Not:

• lejos bien

Yes.

Mi casa can mean either my house or my home, depending on context. In a sentence like this, English could translate it either way:

• The library is really far from my house.
• The library is really far from my home.

Both fit.

Yes, but only if the subject is already clear from context.

For example:

• ¿La biblioteca? Está bien lejos de mi casa.

That works because la biblioteca has already been mentioned. If you say the sentence on its own, it is more natural to keep the subject:

• La biblioteca está bien lejos de mi casa.

Yes. Another common option is quedar:

• La biblioteca queda bien lejos de mi casa.
• La biblioteca queda muy lejos de mi casa.

In Spain, quedar is often used when talking about where something is located relative to another place. Still, estar is completely correct and very common.