Вера важна для неё.

In Russian, the present-tense form of to be is usually omitted.

So:

Вера важна для неё.
literally looks like
Faith/Vera important for her.

But it means Faith is important to her or Vera is important to her, depending on context.

Russian does use быть in the past and future, but not normally in simple present-tense statements like this.

It can be either.

• вера = faith, belief
• Вера = the female name Vera

Because this word is at the beginning of the sentence, it is capitalized anyway, so the capital letter does not help you distinguish the two. Only context or the provided meaning tells you which one is intended.

Grammatically, the sentence works the same either way.

важна is the short form of the adjective важный (important).

Russian often uses the short form of adjectives when the adjective is the main predicate of the sentence:

• Книга важна. = The book is important.
• Работа важна. = The work is important.

Here, важна is feminine singular because вера / Вера is feminine singular.

A quick comparison:

• важный = masculine full form
• важная = feminine full form
• важна = feminine short form

In this sentence, the short form is the normal choice.

неё is the genitive form of она after the preposition для.

The preposition для means for and requires the genitive case:

• для меня = for me
• для тебя = for you
• для неё = for her

So:

• для
  • неё
• для неё = for her

After most prepositions, Russian third-person pronouns add an initial н-.

So you get:

• у неё = at her place / she has
• для неё = for her
• к нему = to him
• с ними = with them

Without a preposition, you can have её. But after для, you need неё.

So:

• Я вижу её. = I see her.
• Это для неё. = This is for her.

Yes, often it can.

For example:

• Ей важна вера.
• Вера важна ей.

These can also mean Faith is important to her.

The difference is mostly about nuance and style:

• ей often sounds more direct: important to her
• для неё can sound a bit more like for her / in her case / in her life

So the given sentence is natural, but it is not the only possible way to express the idea.

The subject is Вера.

That is the thing being described as важна (important).

So the structure is:

• Вера = subject
• важна = predicate adjective
• для неё = prepositional phrase meaning for her

In other words:

[Вера] [важна] [для неё].
[Faith/Vera] [is important] [to/for her].

Yes. Russian word order is flexible, and changing it usually changes emphasis, not the basic meaning.

Possible variants include:

• Вера важна для неё.
• Для неё вера важна.
• Вера для неё важна.

These all mean roughly the same thing, but the focus shifts:

• Вера важна для неё. = neutral
• Для неё вера важна. = emphasizes for her
• Вера для неё важна. = slightly highlights faith/Vera

Russian uses case endings and agreement, so word order is freer than in English.