Я купил новую тетрадь в школу.

Because Russian past-tense verbs agree with the gender of the subject in the singular.

• Я купил = I bought (said by a man)
• Я купила = I bought (said by a woman)

So this sentence sounds as if the speaker is male. In the present tense, Russian verbs do not show gender this way, but in the past tense they do.

Because тетрадь is the direct object of the verb купил, so it has to be in the accusative case.

The base form is:

• новая тетрадь = a new notebook / new notebook (nominative)

But after купил (bought), we need the accusative:

• новую тетрадь

The adjective changes clearly:

• новая → новую

The noun тетрадь happens to stay the same in form here, even though it is also accusative.

Because тетрадь is a feminine noun of the third declension (it ends in -ь), and in the singular its nominative and accusative forms are often identical.

So:

• nominative: тетрадь
• accusative: тетрадь

That is why only the adjective makes the case especially visible here:

• новая тетрадь
• новую тетрадь

This is very common with feminine nouns ending in -ь.

In Russian, nouns ending in -ь can be either masculine or feminine, so you often have to learn the gender with the word.

Тетрадь is feminine, and you can see that from the adjective:

• новая тетрадь
  not
• новый тетрадь

A useful rule of thumb:

• many nouns ending in -ь are feminine
• but some are masculine, so you cannot rely on spelling alone

That is why it is best to memorize new nouns together with an adjective or pronoun, for example:

• эта тетрадь
• новая тетрадь

Because в can take different cases depending on the meaning.

• в школе = in the school / at school → location, so prepositional case
• в школу = to school / for school → direction, destination, or purpose, so accusative case

Here the phrase is в школу, so the noun школа changes to the accusative:

• школа → школу

Even though English often says for school, Russian uses в школу in expressions like this.

In this sentence, it most naturally means for school.

Literally, в школу often means to school:

• Я иду в школу = I’m going to school

But with купить тетрадь, the phrase is understood as showing the purpose:

• Я купил новую тетрадь в школу = I bought a new notebook for school

So yes, the literal form is to school, but the natural meaning here is for school use.

Because Russian has no articles.

So тетрадь can mean:

• a notebook
• the notebook

The exact meaning depends on context.

In this sentence, English might translate it as a new notebook, but Russian does not need a separate word for a.

Russian speakers show definiteness in other ways, such as:

• context
• word order
• intonation
• sometimes words like этот (this) if they want to be more specific

Yes. Russian word order is more flexible than English because case endings show what each word is doing.

The neutral order here is:

• Я купил новую тетрадь в школу.

But other orders are possible, depending on emphasis:

• Новую тетрадь я купил в школу.
  Emphasizes the new notebook
• В школу я купил новую тетрадь.
  Emphasizes for school

Even when the order changes, the cases still tell you the roles:

• новую тетрадь = object
• в школу = destination/purpose phrase

Sometimes, but not as freely as in some other languages.

Russian often omits subjects when the meaning is clear from context, but in the past tense the verb form купил only tells you:

• singular
• masculine

It does not by itself tell you whether the subject is I, you, or he.

So:

• Купил новую тетрадь. could mean I bought a new notebook or he bought a new notebook, depending on context.

That is why я is often kept when needed for clarity.

Yes, but she would say:

• Я купила новую тетрадь в школу.

Only the past-tense verb changes:

• male speaker: купил
• female speaker: купила

The rest of the sentence stays the same.