После университета моя сестра прошла стажировку в банке и потом получила постоянную работу.

Because после requires the genitive case.

• университет = nominative
• университета = genitive

So:

• после школы = after school
• после работы = after work
• после университета = after university

This is just a fixed grammar pattern: whenever you use после, the following noun normally goes into the genitive.

In this sentence, it normally means after finishing university / after graduating from university, not literally after walking out of the building.

Russian often uses после + institution/event in a broad life-stage sense:

• после школы = after school
• после армии = after the army / military service
• после университета = after university

If you wanted to be very explicit, you could say something like после окончания университета (after finishing university), but the original version sounds natural and common.

Because моя сестра is the subject of the sentence, so it stays in the nominative case.

• моя agrees with сестра
• both are feminine singular nominative

So the sentence is structured like:

• моя сестра = my sister
• прошла стажировку = completed an internship
• получила постоянную работу = got a permanent job

Since my sister is the one doing the actions, nominative is the correct case.

Yes, literally пройти often means to go through / pass / complete, and in Russian it is very natural to use it with things like courses, training, exams, and internships.

So прошла стажировку means:

• did an internship
• completed an internship
• went through an internship

This is idiomatic Russian. English and Russian just package the idea differently.

Compare:

• пройти курс = take/complete a course
• пройти обучение = complete training
• пройти стажировку = complete an internship

So even though the literal image is going through something, the real meaning here is successfully doing/completing it.

Yes, you could, but the emphasis changes a little.

• прошла стажировку в банке = she completed an internship at a bank
• стажировалась в банке = she was an intern / did internship training at a bank

The original sentence uses прошла because it highlights the internship as a completed step, which fits well with the next event: and then she got a permanent job.

So:

• стажировалась focuses more on the process
• прошла стажировку focuses more on completing it

Both are past tense, feminine singular forms.

That is because the subject is моя сестра, which is feminine singular.

Base verbs:

• пройти → прошла
• получить → получила

In the Russian past tense, gender matters in the singular:

• masculine: получил
• feminine: получила
• neuter: получило
• plural: получили

So the -а ending tells you the subject is feminine.

Because the sentence describes completed actions in sequence:

1. she completed an internship
2. then she got a permanent job

Perfective verbs are very common when Russian tells a story as a chain of finished events.

• пройти is perfective
• получить is perfective

They answer the question: What happened?

If you used imperfective forms, the focus would be different:

• проходила стажировку = she was doing / used to do an internship
• получала = was receiving / used to receive

That would not fit as naturally here, because the sentence is about concrete, completed milestones.

Because в here means in/at, and with location it takes the prepositional case.

• банк = bank
• в банке = in/at the bank

So:

• в университете = at the university
• в офисе = in the office
• в банке = at the bank

Here it means the place where the internship happened.

Because работу is the direct object of получила, so it goes into the accusative case.

• nominative: постоянная работа
• accusative: постоянную работу

Since работа is a feminine noun, the adjective changes too:

• постоянная → постоянную
• работа → работу

So получила постоянную работу literally means received/got a permanent job.

Not exactly. Постоянная работа most directly means permanent job or regular long-term job, as opposed to something temporary.

It often suggests:

• not temporary
• not just an internship
• a stable position

Depending on context, English might translate it as:

• a permanent job
• a regular job
• sometimes a full-time job

But permanent is the safest match here.

Потом means then / afterwards.

It shows the order of events:

• first she completed the internship
• then she got a permanent job

So it helps the sentence feel like a timeline:

После университета → прошла стажировку → потом получила работу

Russian often uses words like потом, затем, or после этого to connect events in this way.

Yes. Russian word order is fairly flexible, although some orders sound more natural than others.

The original sentence is neutral and natural:

После университета моя сестра прошла стажировку в банке и потом получила постоянную работу.

You could also say:

• Моя сестра после университета прошла стажировку в банке и потом получила постоянную работу.
• После университета моя сестра в банке прошла стажировку и потом получила постоянную работу.

But the original version is probably the clearest and most natural for everyday use.

Russian word order often changes for emphasis, not basic grammar. The case endings already show what each word is doing.

Because Russian has no articles.

So университета, банке, and работу can mean:

• the university / a university
• the bank / a bank
• the job / a job

The exact meaning comes from context.

In this sentence:

• в банке is naturally understood as at a bank
• получила постоянную работу is naturally understood as got a permanent job

English needs articles, but Russian does not.