Перед зачётом брат написал шпаргалку, но потом решил ею не пользоваться.

Because the preposition перед normally takes the instrumental case when it means before in time or in front of in space.

So:

• зачёт = nominative
• зачётом = instrumental singular

In this sentence, перед зачётом means before the test / before the pass-fail assessment.

A useful comparison:

• перед зачётом = before the test, often with a sense of right before / in the period leading up to it
• до зачёта = before the test / until the test, often more neutral

Both can work in many contexts, but перед is very natural here.

Not exactly.

In Russian academic contexts, зачёт is usually a pass/fail assessment or a course requirement that you either pass or do not pass. It is often less formal or less extensive than an экзамен, though that depends on the institution.

Very roughly:

• зачёт = pass/fail credit, check, assessment
• экзамен = exam, usually graded

So a learner should not automatically translate зачёт as exam in every situation.

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

Here the idea is that the brother finished writing the cheat sheet. The result exists.

• написал шпаргалку = he wrote a cheat sheet, and it was completed
• писал шпаргалку = he was writing a cheat sheet / used to write cheat sheets / spent time writing one

Since the sentence talks about a single completed action before the test, написал is the natural choice.

Шпаргалка means a cheat sheet, crib note, or a small hidden note used to help someone during a test.

It is a very common Russian word, and it usually has a slightly informal, everyday feel. In context, it strongly suggests something a student might secretly use during a test.

Because шпаргалку is the accusative singular, and it is the direct object of написал.

The verb написать answers the question what?:

• написал что? → шпаргалку

Since шпаргалка is a feminine noun ending in -а, its accusative singular changes to -у:

• nominative: шпаргалка
• accusative: шпаргалку

Because the verb пользоваться requires the instrumental case, not the accusative.

So the pronoun referring to шпаргалка must also be in the instrumental:

• она = nominative
• её = accusative/genitive
• ею = instrumental

Here:

• пользоваться чем? → ею

So ею means with it / by means of it, which is how Russian expresses use it with this verb.

That is simply the case government of the verb. In Russian, many verbs require a specific case, and you have to learn them together with the verb.

For пользоваться, the pattern is:

• пользоваться чем?
  • instrumental

Examples:

• пользоваться словарём = to use a dictionary
• пользоваться телефоном = to use a phone
• пользоваться шпаргалкой = to use a cheat sheet

A useful comparison:

• пользоваться чем?
  • instrumental
• использовать что?
  • accusative

So Russian has two common ways to say use, but they govern different cases.

Because пользоваться here presents the action in a more general or ongoing way: to use it, to rely on it, to make use of it during the test.

That works well because using a cheat sheet is not just a single instant; it can be seen as a continuing activity.

You could also say:

• решил ею не воспользоваться

That would sound more like decided not to make use of it on that particular occasion, with stronger one-time perfective focus.

So:

• не пользоваться = not to use it, not to be using it
• не воспользоваться = not to make use of it once / on this occasion

In this sentence, the imperfective sounds very natural.

Yes. In modern Russian, the instrumental singular of она can be either:

• ею
• ей

So both are possible:

• решил ею не пользоваться
• решил ей не пользоваться

The difference is mostly stylistic:

• ею often sounds a bit clearer, slightly more careful, sometimes a little more literary
• ей is very common in everyday speech

After a preposition, however, you normally get ней, not ей/ею:

• с ней
• перед ней

No. Russian word order is flexible.

These are all possible:

• решил ею не пользоваться
• решил не пользоваться ею
• потом решил ею не пользоваться

The sentence you have is natural because ею is placed before the infinitive and smoothly connects back to шпаргалку.

One important point: не belongs with the verb пользоваться, so the negative meaning is not use. The pronoun can move around more freely than the не.

Because но is a coordinating conjunction meaning but, and it joins two clauses:

• Перед зачётом брат написал шпаргалку
• но потом решил ею не пользоваться

In Russian, a comma is normally placed before но when it connects two clauses like this.

So the punctuation is standard and expected.