Я села на лавочку у озера и открыла книгу.

Because the past tense in Russian agrees with the speaker’s gender.

• села = I sat down said by a female
• сел = I sat down said by a male

The same applies to открыла:

• открыла = said by a female
• открыл = said by a male

So this sentence is spoken by a woman or girl.

These verbs mean different things.

• села comes from сесть and means sat down
• сидела comes from сидеть and means was sitting / sat

So:

• Я села на лавочку = I sat down on the bench
• Я сидела на лавочке = I was sitting on the bench

In your sentence, the speaker is describing a sequence of completed actions: first she sat down, then she opened a book.

Because открыла is perfective, and it presents the action as completed.

• открыла = opened as a finished act
• открывала = was opening / used to open / opened repeatedly, depending on context

In this sentence, the speaker is telling what happened next in the story, so the perfective form is natural:

• села → sat down
• открыла → opened

Both actions are seen as single completed events.

Because after на with movement toward something, Russian uses the accusative case.

• dictionary form: лавочка
• accusative singular: лавочку

So:

• на лавочку = onto a bench / to a bench
• на лавочке = on a bench as a location

This sentence uses на лавочку because the action is sitting down onto the bench.

This is a very common Russian pattern:

• на + accusative = motion toward a surface
• на + prepositional = location on a surface

So:

• Я села на лавочку = I sat down on the bench
  movement to the bench
• Я сидела на лавочке = I was sitting on the bench
  location on the bench

The same kind of contrast appears in many places:

• положить на стол = put onto the table
• лежать на столе = lie on the table

Because у normally takes the genitive case and often means by, near, beside.

• dictionary form: озеро = lake
• genitive singular: озера

So:

• у озера = by the lake / near the lake

This is a very common pattern:

• у дома = by the house
• у окна = by the window
• у реки = by the river

Because книгу is the direct object of открыла, so it goes into the accusative case.

• dictionary form: книга
• accusative singular: книгу

So:

• открыла книгу = opened a book

For many feminine nouns ending in -а or -я, the accusative singular changes:

• книга → книгу
• газета → газету
• неделя → неделю

It can sometimes be dropped, but here Я is helpful.

In the present tense, Russian often omits the subject because the verb ending shows the person clearly:

• Читаю = I am reading

But in the past tense, forms like села and открыла show gender, not person. So without Я, the sentence could mean something like she sat down and opened if the context did not already make the subject clear.

So:

• Я села на лавочку и открыла книгу = clear and natural
• Села на лавочку и открыла книгу = possible in context, but less explicit

Because the same subject continues for both verbs, and Russian usually does not repeat it unless there is a reason to emphasize it.

So this is natural:

• Я села на лавочку у озера и открыла книгу.

Repeating я would sound heavier:

• Я села на лавочку у озера, и я открыла книгу.

That version is possible, but usually only if you want emphasis, contrast, or a special rhythm.

They are close in meaning, but there is a slight nuance.

• лавочка often feels a bit more everyday, cozy, or diminutive
• скамейка is also a normal word for bench, often a bit more neutral

In many contexts, both can be translated simply as bench.

Here, лавочка fits very naturally for a small bench by a lake, especially in a calm or story-like sentence.

Yes, Russian word order is fairly flexible.

The sentence as written is a neutral, natural order:

• Я села на лавочку у озера и открыла книгу.

But Russian can move parts around for emphasis:

• У озера я села на лавочку и открыла книгу.
  emphasizes by the lake
• Книгу я открыла, сев на лавочку у озера.
  emphasizes the book

Even though Russian allows changes, the original sentence is probably the most straightforward version for a learner.

Yes. Both verbs are perfective:

• сесть → села
• открыть → открыла

Perfective verbs are commonly used in narration when you list completed actions one after another.

That gives the sentence a clear storyline:

1. she sat down
2. she opened the book

If you used imperfective verbs instead, the feeling would change. It might sound more like background description, repetition, or an ongoing process rather than a simple sequence of completed events.

Russian has no articles like a and the, so книгу can mean either a book or the book, depending on context.

So:

• открыла книгу could be opened a book
• or opened the book

The surrounding context tells you which one is meant. If this sentence appears by itself, English often chooses a book because it sounds more natural when introducing the object for the first time.