Она бережно положила старые письма в коробку и закрыла её.

Because the subject is она (she), and these are past-tense verbs in the feminine singular.

In Russian past tense:

• masculine: usually -л
• feminine: -ла
• neuter: -ло
• plural: -ли

So:

• она положила = she put
• она закрыла = she closed

If the subject were он, it would be:

• Он бережно положил старые письма в коробку и закрыл её.

Because бережно is an adverb, not an adjective.

It describes how she put the letters down: carefully / gently.

Adverbs in Russian do not change for gender, number, or case.
So бережно stays the same no matter who is doing the action:

• Она бережно положила...
• Он бережно положил...
• Они бережно положили...

Here старые письма is the direct object of положила.

The noun письма is plural of письмо (letter), and it is inanimate. In Russian, for inanimate plural nouns, the accusative plural looks the same as the nominative plural.

So:

• nominative plural: старые письма
• accusative plural: старые письма

That is why the phrase stays старые письма here.

Compare with animate nouns, where accusative plural is different:

• Я вижу старых друзей. = I see old friends.

Because this sentence shows motion into the box, not location inside it.

With в:

• в + accusative = into somewhere (direction/movement)
• в + prepositional = in somewhere (location)

So:

• положила в коробку = put into the box
• письма лежат в коробке = the letters are in the box

This is a very common Russian pattern:

• идти в школу = go to school
• быть в школе = be at school

Because её is the correct form of the pronoun meaning her or it in this context, and it refers back to коробку.

The noun коробка (box) is grammatically feminine, so when Russian refers back to it, it uses the feminine pronoun:

• закрыла её = closed it

Even though box is not feminine in English, Russian nouns have grammatical gender, and pronouns must agree with that gender.

Grammatically, the form is the same, but in this sentence it means it, because it refers to коробку (box), not to a female person.

So:

• Я вижу её. could mean I see her or I see it
• You understand which one it means from context

Here the context makes it clear:

• закрыла её = closed it (the box)

Both verbs are perfective:

• положила from положить
• закрыла from закрыть

Perfective verbs present actions as completed whole events. That fits this sentence because it describes a completed sequence:

1. she put the letters into the box
2. she closed it

If you used imperfective forms, the meaning would change and might sound like ongoing, repeated, or process-focused action:

• клала instead of положила
• закрывала instead of закрыла

So the perfective is natural here because the sentence tells what happened and was finished.

Because положила is the perfective verb for a single completed act of placing something.

Very roughly:

• класть / клала = to be placing, to place repeatedly, or to focus on the process
• положить / положила = to put, to place down, successfully and completely

In a narrative sentence about one finished action, положила is the normal choice.

For example:

• Она клала письма в коробку, когда я вошёл.
  She was putting letters into the box when I came in.
• Она положила письма в коробку и закрыла её.
  She put the letters into the box and closed it.

Because и is simply joining two verbs with the same subject:

• положила
• закрыла

In English, the structure is similar:
She carefully put the old letters into the box and closed it.

Russian normally does not put a comma before и when it connects two homogeneous predicates belonging to one subject.

A comma would be needed in other situations, for example if there were two separate clauses with more independence or extra inserted material.

Yes. Russian word order is more flexible than English word order.

The sentence:

• Она бережно положила старые письма в коробку и закрыла её.

is natural and neutral. But other orders are also possible, with different emphasis:

• Она положила старые письма в коробку бережно...
  possible, but the adverb sounds more highlighted or stylistically marked
• Старые письма она бережно положила в коробку...
  emphasizes the old letters
• В коробку она бережно положила старые письма...
  emphasizes into the box

So the basic meaning stays the same, but information focus changes.

Because it agrees with письма, which is plural.

Russian adjectives must agree with the noun they describe in:

• gender
• number
• case

Here:

• письма = plural
• direct object, accusative plural
• inanimate

So the adjective is also plural accusative, which for inanimate nouns looks like nominative plural:

• старые письма

If it were singular feminine, for example:

• старую книгу = an old book

It is pronounced roughly like yi-YO, with the stress on the second syllable.

A more precise approximation is:

• её ≈ [jɪˈjo]

Important points:

• the written ё always indicates the yo sound
• the stress is on ё
• in normal writing, Russians sometimes replace ё with е, so you may also see ее, but here the pronunciation is still её

So in speech:

• закрыла её sounds roughly like za-KRY-la yi-YO

Sometimes Russian can omit objects if they are obvious from context, but here закрыла её sounds more complete and natural.

If you said:

• Она бережно положила старые письма в коробку и закрыла.

many listeners would feel that something is missing, because закрыть usually needs an object: closed what?

So keeping её is the normal choice.

Because Russian grammatical gender is a property of nouns, not of whether something is alive.

• коробка ends in -а, so it is feminine
• therefore pronouns and modifiers referring to it are feminine too

This is very normal in Russian:

• книга → feminine
• стол → masculine
• окно → neuter

So её refers to коробку simply because коробка is a feminine noun.