Она не дала ему договорить и сразу задала другой вопрос.

In this pattern, дать + dative + infinitive often means to let / allow someone to do something.

So:

• Она не дала ему договорить = She didn’t let him finish speaking

Literally, it is something like She didn’t give him [the chance/time] to finish speaking, but in natural English we translate it as didn’t let him finish.

This is a very common Russian structure:

• Дай мне сказать. = Let me speak.
• Она не дала ему уйти. = She didn’t let him leave.

Because with дать in this meaning, the person who is allowed or not allowed to do something is put in the dative.

So:

• ему = to him
• Она не дала ему договорить literally has the structure She did not give to him to finish speaking

That literal wording sounds strange in English, but grammatically it helps explain the case choice.

Договорить means to finish saying something or to finish speaking.

It is built from:

• говорить = to speak / say
• prefix до- = often adds the idea of doing something to the end / finishing it

So:

• говорить = to speak
• договорить = to finish speaking / say the rest

In this sentence, it implies that he had already started speaking, but she interrupted him before he could finish.

Because it depends on дала in the pattern не дала ему + infinitive.

Russian often uses an infinitive after verbs meaning let, make, help, begin, want, can, etc.

Here the structure is:

• не дала ему договорить = didn’t let him finish speaking

You are not saying he finished speaking. You are saying she did not allow the action of finishing speaking.

Because the subject is она (she).

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

• он дал / задал
• она дала / задала
• оно дало / задало
• они дали / задали

So since the subject is она, both verbs take the feminine singular past form:

• дала
• задала

Because the sentence describes two completed, single actions in sequence:

1. she didn’t let him finish
2. she immediately asked another question

Perfective verbs are very natural here because the speaker is presenting the events as whole, completed actions.

Compare:

• не дала = she didn’t let him (on that occasion, as a completed event)
• задала = she asked/posed (one completed question)

If you used imperfective forms like не давала or задавала, it could suggest repetition, duration, or background description instead.

In Russian, задать вопрос is the standard collocation for to ask a question.

So:

• задать вопрос = to ask a question

Russian does also have спросить, but it is usually used differently:

• спросить что-то = to ask something
• спросить кого-то о чём-то = to ask someone about something

A useful distinction is:

• задать вопрос focuses on posing a question
• спросить is the more general verb to ask

In this sentence, задала другой вопрос sounds very natural and idiomatic.

Другой can mean all of these depending on context:

• another
• different
• other

Here задала другой вопрос most naturally means:

• asked another question
• or asked a different question

The idea is that instead of letting him finish, she moved on and asked a new question.

Also note that другой does not usually mean specifically second. If you wanted the second question, that would be второй вопрос.

It actually is in the accusative case, but for a masculine inanimate noun like вопрос, the accusative singular looks the same as the nominative singular.

So:

• nominative: вопрос
• accusative: вопрос

That is why the form does not visibly change.

And другой also appears as:

• nominative masculine singular: другой
• accusative masculine singular inanimate: другой

So другой вопрос is the expected form after задала.

Because this is one sentence with one subject (она) and two coordinated verbs:

• не дала
• задала

Russian normally does not put a comma before и when it simply joins two homogeneous parts of the sentence, such as two verbs with the same subject.

So:

• Она не дала ему договорить и сразу задала другой вопрос.

No comma is needed.

Yes, сразу means immediately, right away, or straight away.

It shows that the second action happened without delay:

• she didn’t let him finish speaking
• and immediately asked another question

So it strengthens the feeling of interruption and quick continuation.

The word order is natural and neutral, but Russian word order is fairly flexible.

This sentence is structured as:

• Она — topic/subject
• не дала ему договорить — first action
• и сразу задала другой вопрос — second action

Putting сразу before задала emphasizes that the next action happened at once.

Other word orders are possible, but they would slightly change emphasis. For example:

• Она сразу задала другой вопрос и не дала ему договорить
  This would foreground the fact that she immediately asked another question.

The original version is very natural for telling the events in order.