Я вижу её новую книгу на столе.

Because вижу (I see) takes a direct object, and direct objects usually go in the accusative case.

The dictionary form is:

• новая книга = a new book / new book
  
  • новая = feminine singular nominative
  • книга = feminine singular nominative

But after вижу, both words change to accusative:

• новую книгу

For feminine singular nouns and adjectives, this is very common:

• новая книга → новую книгу
• интересная статья → интересную статью

So in this sentence:

• Я вижу её новую книгу = I see her new book

Because Russian uses different cases after на depending on whether you mean:

• location = where something is
• motion toward a place = where something is going

Here:

• на столе uses the prepositional case
• it means on the table as a location

Compare:

• Книга на столе. = The book is on the table.
• Я кладу книгу на стол. = I put the book onto the table.

So:

• на столе = on the table, located there
• на стол = onto the table, movement toward it

In your sentence, the book is already there, so на столе is correct.

Here её means her in the possessive sense:

• её книга = her book

So:

• её новую книгу = her new book

A useful thing to know is that её does not change form here. Unlike many other Russian words, this possessive form stays the same:

• её книга
• её книгу
• её книги

It always remains её.

Also, её can sometimes also mean her as an object pronoun:

• Я вижу её. = I see her.

So Russian learners often ask how to tell the difference. The answer is: context.

• Я вижу её. = I see her.
• Я вижу её книгу. = I see her book.

In your sentence, because её is followed by новую книгу, it clearly means her as a possessive.

In Russian, possessive words like мой, твой, его, её, наш often come before descriptive adjectives:

• её новая книга = her new book
• мой старый дом = my old house

So её новую книгу is the normal, neutral order.

You can sometimes change word order in Russian for emphasis, but the neutral version is:

• её новую книгу

That is the most natural way to say her new book in an ordinary sentence.

Yes, it could.

Russian often omits subject pronouns when the verb already makes the subject clear.

• Я вижу = I see
• Вижу also clearly means I see from the verb ending -у

So both are possible:

• Я вижу её новую книгу на столе.
• Вижу её новую книгу на столе.

The version with Я is a bit more explicit and can add emphasis, contrast, or clarity. For example:

• Я вижу её новую книгу, а он не видит.
  I see her new book, but he doesn’t.

So Я is not always necessary, but it is perfectly normal.

Because видеть and смотреть are different verbs.

• видеть = to see
• смотреть = to look / to watch

So:

• Я вижу книгу. = I see the book.
• Я смотрю на книгу. = I am looking at the book.

In your sentence, the idea is simply that the speaker perceives the book, so вижу is the right choice.

Also notice the grammar difference:

• видеть usually takes a direct object in the accusative:
  • вижу книгу
• смотреть usually uses на
  • accusative:
    • смотрю на книгу

So you say:

• Я вижу её новую книгу на столе. not
• Я смотрю её новую книгу... ❌

In normal understanding, на столе tells you where the book is:

• I see her new book, which is on the table.

That is the most natural reading.

Russian often allows this kind of phrase to attach quite naturally to the noun being talked about, especially when it makes obvious real-world sense.

If you wanted to say I see her new book while sitting at the table, Russian would usually express that differently, for example with:

• за столом = at the table

So:

• Я вижу её новую книгу на столе. = I see her new book on the table.
• Я вижу её новую книгу, сидя за столом. = I see her new book while sitting at the table.

Because in Russian, the direct object of many verbs goes in the accusative whether it is animate or inanimate.

What changes is how the accusative looks.

For inanimate feminine singular nouns, the accusative often has its own visible ending:

• книга → книгу
• машина → машину

For inanimate masculine singular nouns, accusative often looks the same as nominative:

• стол → стол
• дом → дом

So the fact that книгу is not a person does not matter here. It is still the direct object of вижу, so it goes into the accusative.

No, Russian word order is more flexible than English word order.

The neutral order here is:

• Я вижу её новую книгу на столе.

But Russian can move things around for emphasis or style, for example:

• Её новую книгу я вижу на столе.
  Emphasizes her new book
• На столе я вижу её новую книгу.
  Emphasizes on the table

Even when the order changes, the case endings still show the grammatical relationships:

• книгу is still the object
• на столе is still a location phrase

So word order matters for focus and emphasis, but not as rigidly as in English.

Столе is the prepositional singular form of стол (table).

The dictionary form is:

• стол

After на when expressing location, Russian often uses the prepositional case:

• на столе = on the table
• в доме = in the house
• в книге = in the book

So in your sentence:

• на столе = on the table

This is a very common pattern worth memorizing:

• в / на + prepositional for location

Yes. In ordinary writing, её is often written as ее.

So these mean the same thing:

• её новую книгу
• ее новую книгу

The difference is only the dots over ё:

• ё shows the pronunciation more clearly
• е is often used instead in everyday printed text

For learners, it is very helpful to remember that the standard pronunciation here is with ё:

• её pronounced roughly like yi-YO

So if you see ее, you should often mentally recognize it as её.

Here are the dictionary forms:

• Я = I
• вижу → dictionary form видеть = to see
• её = her
• новую → dictionary form новый = new
• книгу → dictionary form книга = book
• на = on
• столе → dictionary form стол = table

This is a useful habit in Russian: when you see a sentence, try to identify which words are changed by case or agreement and what their base forms are.