Depois da fotocópia, vou rever o índice do livro.

Because depois normally takes de when it means after and is followed by a noun or a verb.

Here the noun is a fotocópia, so:

• depois de + a fotocópia → depois da fotocópia

This is just a normal contraction:

• de + a = da
• de + o = do

So depois da fotocópia means after the photocopy or after making the photocopy, depending on context.

Also note:

• depois by itself = later / afterwards
• depois de = after

Fotocópia usually means photocopy.

In this sentence, it can be understood in two slightly different ways:

• the photocopy itself: after the photocopy
• the act of photocopying / making a photocopy: after making the photocopy

In everyday use, Portuguese often allows this kind of shorthand. If you want to make the action very explicit, you could say:

• Depois de fazer a fotocópia...
• Depois de tirar uma fotocópia...

Both make it clearer that you mean the action.

Vou rever is the very common near-future structure in Portuguese:

• ir + infinitive

So:

• vou rever = I’m going to review / I will review

This is extremely common in spoken Portuguese, including in Portugal. It often sounds more natural and everyday than the simple future:

• reverei

The simple future is grammatical, but it can sound more formal, more written, or less conversational in many contexts.

Yes. Rever is related to ver.

• ver = to see
• rever = to see again, review, go over again, revise

In this sentence, rever is best understood as:

• to review
• to look over again

So vou rever o índice do livro means that the speaker is going to look at the book’s index/contents again.

Because it comes from ver, some of its conjugation patterns are related to ver, but here it is in the infinitive, so you do not have to worry about that yet.

It can mean either, depending on context.

With books, índice often refers to:

• the table of contents
• the index
• more generally, the contents listing / structured list of sections

Context decides which meaning is most likely.

In many everyday situations, if someone says o índice do livro, they may mean the section that helps you see how the book is organized. If you want to be very specific, context or a different word may help.

Portuguese uses definite articles much more often than English.

So where English might say:

• review the book index
• review the index of the book

Portuguese naturally says:

• rever o índice do livro

Here:

• o índice = the index
• do livro = of the book

And again:

• de + o = do

This article use is very normal in Portuguese, and English speakers often need time to get used to it.

Yes, absolutely.

That version is often clearer because it explicitly shows the action:

• Depois de fazer a fotocópia, vou rever o índice do livro.

You may also hear:

• Depois de tirar uma fotocópia...

These are all natural, but they have slightly different feels:

• Depois da fotocópia... = shorter, more compact
• Depois de fazer a fotocópia... = clearer about the action
• Depois de tirar uma fotocópia... = also very natural in everyday usage

It is not absolutely required, but it is very natural.

Depois da fotocópia is an introductory time expression. In Portuguese, a comma after that kind of opening phrase is common, especially if the writer wants a clear pause.

So both are possible:

• Depois da fotocópia, vou rever o índice do livro.
• Depois da fotocópia vou rever o índice do livro.

The version with the comma is a little easier to read and sounds slightly more structured.

They mainly help show stress and, in some cases, vowel quality.

• fotocópia: the stress falls on có
• índice: the stress falls on the first syllable, ín

So roughly:

• fo-to-CÓ-pia
• ÍN-di-ce

For an English speaker, that is useful because Portuguese stress is not always predictable just from spelling.

In European Portuguese, unstressed vowels are often reduced more than in Brazilian Portuguese, so the actual sound may be less open and less fully pronounced than an English speaker expects.

Yes, it is natural and grammatical in European Portuguese.

That said, depending on nuance, a speaker might also say:

• Depois da fotocópia, vou ver o índice do livro.
• Depois da fotocópia, vou consultar o índice do livro.
• Depois da fotocópia, vou dar uma vista de olhos ao índice do livro.

These alternatives change the tone slightly:

• ver = simply look at
• consultar = consult/check
• dar uma vista de olhos = have a quick look
• rever = review / look over again

So the original sentence sounds especially natural if the speaker has already looked at the index before and plans to check it again.