После неудачного зачёта у него будет пересдача в конце месяца, а я на пересдачу так и не пошла, потому что смогла сдать всё раньше.

It is in the genitive case because the preposition после always takes genitive.

So:

• после = after
• неудачный зачёт = an unsuccessful credit/test
• после неудачного зачёта = after an unsuccessful credit/test

The endings show genitive singular masculine:

• неудачный → неудачного
• зачёт → зачёта

This is a very common pattern:

• после урока = after the lesson
• после экзамена = after the exam
• после трудного дня = after a difficult day

Not quite. Зачёт is a specific educational term, especially common in Russian-speaking universities.

Very roughly:

• экзамен = an exam, usually graded
• зачёт = a pass/fail assessment, often less formal than an exam

A зачёт may involve:

• answering a few questions,
• showing completed coursework,
• proving you know the material well enough to get a pass.

So in many contexts, зачёт is better understood as a course credit / pass-fail assessment, not a full exam.

Here неудачный means something like unsuccessful or failed.

So:

• неудачный зачёт = an unsuccessful attempt at the credit/test

Using плохой would mean bad, but неудачный focuses more on the outcome not going well.

Compare:

• плохой ответ = a bad answer
• неудачная попытка = an unsuccessful attempt

In this sentence, неудачный sounds natural because the person did not pass.

Пересдача means a retake — usually of an exam, test, or зачёт.

It comes from the verb idea пересдать = to retake and pass / to take again.

Examples:

• У меня пересдача. = I have a retake.
• Идти на пересдачу. = to go to a retake.
• Сдать с пересдачи is less standard than simply saying сдать на пересдаче = to pass on the retake.

In your sentence, у него будет пересдача means he will have to take it again later.

Because Russian usually does not use иметь the way English uses to have.

Instead, Russian often uses the structure:

• у + genitive + быть

So:

• у него будет пересдача = literally at him there will be a retake
• natural English: he will have a retake

This pattern is extremely common:

• У меня будет экзамен. = I will have an exam.
• У неё будет встреча. = She will have a meeting.

So this is normal Russian, even if it looks unusual from an English point of view.

This expression has two parts:

• в конце = at the end
• месяца = of the month

More precisely:

• в
  • конце: here конце is prepositional, in the fixed phrase в конце = at the end
• месяца is genitive because it depends on конец: the end of the month

So literally it is:

• in the end of the month

but in natural English:

• at the end of the month

You will see the same pattern in:

• в конце недели = at the end of the week
• в конце года = at the end of the year

Because пойти / идти на something expresses movement to an event or place, and that usually takes the accusative.

So:

• идти на лекцию = to go to a lecture
• пойти на экзамен = to go to an exam
• пойти на пересдачу = to go to a retake

Here:

• пересдача → на пересдачу (accusative singular)

Compare:

• Я пошла на пересдачу. = I went to the retake.
• Я была на пересдаче. = I was at the retake.

The first is motion toward it; the second is location/state.

Так и не is a very common expression that adds the meaning of in the end, not or never actually did.

So:

• я на пересдачу так и не пошла

means more than just I didn’t go to the retake. It suggests something like:

• I never ended up going to the retake
• I didn’t go after all
• I just never went

This often implies there was some expectation that the action might happen, but it never did.

Examples:

• Он так и не позвонил. = He never did call.
• Мы так и не встретились. = We never ended up meeting.

Because the speaker is female.

In the past tense, Russian verbs agree in gender and number:

• masculine: пошёл, смог
• feminine: пошла, смогла
• neuter: пошло, смогло
• plural: пошли, смогли

Since the speaker says я ... не пошла and смогла, we know the speaker is a woman.

If a man were speaking, it would be:

• а я на пересдачу так и не пошёл, потому что смог сдать всё раньше.

This is about both meaning and aspect.

смочь means to manage / to be able successfully, and it usually points to a completed result.
So:

• смогла сдать = she managed to pass / managed to get everything done

The infinitive сдать is also perfective, which fits the idea of a completed achievement.

Compare:

• могла сдавать = was able to be taking / could be taking
  This sounds more about general possibility or process.
• смогла сдать = succeeded in passing / succeeded in getting it done

In this sentence, the speaker means she completed everything earlier, so the perfective pair is natural.

Here сдать всё раньше means something like:

• to pass/complete everything earlier
• to get all of it done before the retake

The word всё is neuter singular and means everything.

Compare:

• всё = everything
• все = everyone / all people, or all items in the plural

So here всё is correct because it refers to all the required things as a whole — perhaps all assignments, tests, requirements, or parts of the course.

Examples:

• Я сдала всё. = I passed/completed everything.
• Все сдали зачёт. = Everyone passed the credit/test.

Yes — сдать has several meanings, and that often confuses learners.

Common meanings include:

• to hand in / submit
• to сдавать экзамен / зачёт = to take an exam/test
• to сдать экзамен / зачёт = to pass an exam/test
• to rent out
• to give up / turn over

In academic contexts, сдать often means to pass or to complete successfully.

So here:

• смогла сдать всё раньше means she managed to pass/complete everything earlier, not just physically hand something in.

Here раньше means earlier / beforehand, with the comparison understood from context.

The implied meaning is:

• earlier than the retake,
• earlier than expected,
• before she needed to go to the retake.

So the logic is:

• he has a retake at the end of the month,
• but she did not go to the retake,
• because she managed to pass everything earlier.

Russian often leaves this comparison implicit when it is obvious from context.

Yes, Russian word order is flexible, but the chosen order helps shape emphasis.

In:

• а я на пересдачу так и не пошла

the phrase на пересдачу is placed before the verb to highlight the destination/event: as for the retake, I never went.

Other possible versions are grammatically possible, for example:

• а я так и не пошла на пересдачу
• а я не пошла на пересдачу

But the original version gives slightly stronger focus to на пересдачу.

Also, так и не naturally sits close to the verb, because it modifies the whole action.

Both а and но can be translated as but, but they are not the same.

Here а works well because it sets up a contrast between two situations:

• у него будет пересдача...
• а я ... не пошла

So it feels like:

• whereas
• while
• and as for me

Но would sound more like a stronger contradiction or objection.

Very roughly:

• а = contrast/comparison
• но = but/however, stronger opposition

So а is a very natural choice here because the speaker is comparing his situation with hers.