Я ждала подругу полчаса у входа, а потом сама зашла в кафе.

Because the speaker is female.

In the Russian past tense, singular verbs agree with the subject in gender:

• ждал = a man was waiting
• ждала = a woman was waiting
• ждало = it was waiting
• ждали = they were waiting

So Я ждала tells you that the person speaking is a woman.

Because подруга is the direct object of ждала, so it goes into the accusative case.

• nominative: подруга = female friend
• accusative: подругу

With verbs like ждать (to wait for), Russian uses a direct object:

• Я ждала подругу = I was waiting for my female friend

This is different from English, where for is required. In Russian, ждать usually takes the object directly.

Yes. Подруга means a female friend.

Compare:

• друг = male friend
• подруга = female friend

So Я ждала подругу specifically means the speaker was waiting for a woman friend, not just any friend of unspecified gender.

Полчаса means half an hour.

In this sentence it expresses duration: how long the speaker was waiting. Russian often uses the accusative for length of time.

• ждать полчаса = to wait for half an hour
• работать весь день = to work all day
• спать всю ночь = to sleep all night

So полчаса here answers the question for how long?

Also note that полчаса is a fixed form meaning half an hour.

Входа is in the genitive case.

The preposition у normally requires the genitive and often means:

• by
• near
• at

So:

• у входа = by the entrance / near the entrance

Compare:

• вход = entrance
• у входа = by the entrance

Because у входа is the most natural way to say by the entrance / near the entrance.

• у входа = by/near the entrance
• на входе can sometimes mean at the entrance too, but often sounds more like at the entrance area or is used in specific contexts
• в входе is not correct here

So in this sentence, у входа is the normal and idiomatic choice.

А often links two clauses with a sense of contrast, transition, or change of situation.

Here the idea is:

• first: she waited for her friend
• then: she changed her plan and went in herself

So а потом feels natural because it marks that shift:

• Я ждала подругу полчаса у входа, а потом... = I waited for my friend for half an hour at the entrance, and then...

Using и потом would sound more like simple addition. А потом better matches the idea of but then / and then after that.

Сама here means something like:

• myself
• on my own
• by myself

It adds emphasis: the friend did not come, so the speaker went into the café alone.

So:

• потом сама зашла в кафе = then I went into the café myself / on my own

Because the speaker is female, it is сама. A male speaker would say сам.

Both зайти and войти can mean to go in / enter, but зайти often sounds more natural in everyday situations like going into a café, shop, or room.

• зайти в кафе = to go into a café
• войти в кафе = to enter a café

Войти can sound a bit more formal or focused on the act of entering itself.
Зайти is very common in everyday speech.

So сама зашла в кафе sounds natural and conversational.

Because зашла is perfective. It presents the action as a completed whole: she ended up going into the café.

• зашла = went in, entered
• заходила = was going in / used to go in / dropped in repeatedly, depending on context

In this sentence, the sequence is clear:

1. she waited
2. then she went in

So the completed action зашла is the right choice.

With verbs of motion, в often takes the accusative to mean into:

• в дом = into the house
• в школу = into the school

Here, зайти в кафе means to go into the café, so it is accusative.

However, кафе is an indeclinable noun, so its form does not change. That is why it still looks like кафе.

So:

• в кафе can mean in the café or into the café
• the meaning depends on the verb and context

Here, because of зашла, it means into the café.

No, Russian word order is fairly flexible, but the given order sounds natural.

The sentence:

• Я ждала подругу полчаса у входа, а потом сама зашла в кафе.

is neutral and clear.

Russian can move things around for emphasis, for example:

• Полчаса я ждала подругу у входа...
  Emphasis on the duration.
• У входа я ждала подругу полчаса...
  Emphasis on the location.
• А потом я сама зашла в кафе.
  Slightly more emphasis on I.

So the word order can change, but each version shifts the focus a little.

Yes, depending on context, Я ждала can be translated in different past-time ways in English:

• I waited
• I was waiting
• I had been waiting

Russian past tense does not directly match English tense labels one-to-one. The form ждала is imperfective past, so it often describes an ongoing or background action.

Because the sentence includes полчаса and then a next action, English often prefers:

• I waited for my friend for half an hour... or
• I had been waiting for my friend for half an hour...

The exact English version depends on the style of translation.