Какие бы вопросы ни возникли на семинаре, преподаватель ответит на них без спешки.

The combination какие бы ... ни is a common Russian concessive pattern meaning whatever / no matter what kind of.

So:

• Какие бы вопросы ни возникли... = Whatever questions may arise...
• More literally: No matter what questions arise...

This structure often includes:

• a question word: кто, что, какой, где, когда, как etc.
• бы
• ни
• a verb

Examples:

• Кто бы ни пришёл, я поговорю с ним. = Whoever comes, I’ll speak with him.
• Что бы ни случилось, сохраняй спокойствие. = Whatever happens, stay calm.

So here, бы helps create that indefinite / no matter which meaning.

In this sentence, ни is not a normal negation like не. It is part of the fixed concessive structure кто бы ни... / что бы ни... / какие бы ни...

So in:

• Какие бы вопросы ни возникли...

the ни does not mean not by itself. Instead, it helps express regardless of what or no matter which.

This is different from:

• не = standard negation
• ни here = part of a set pattern used for emphasis and generality

A learner should think of бы ... ни as a unit in this kind of sentence.

Because вопросы is:

• plural
• nominative case here

So the adjective/pronoun must agree with it:

• masculine singular: какой вопрос
• feminine singular: какая книга
• neuter singular: какое слово
• plural: какие вопросы

In this sentence, вопросы is the subject of возникли, so nominative plural is needed:

• Какие бы вопросы ни возникли...

Because вопросы is the grammatical subject of the verb возникли.

The clause means:

• questions arose / come up

In Russian, the thing that arises is the subject, so it appears in the nominative:

• вопросы возникли = questions arose

That is why it is:

• какие... вопросы = nominative plural

not, for example, accusative or genitive.

Возникли is plural because its subject is вопросы, which is plural.

As for the tense: in this construction, Russian often uses the past tense form after бы ... ни to express something like whatever may arise / whatever might come up. This is very natural Russian usage.

So although English may use present or future-like wording:

• Whatever questions arise...
• Whatever questions may come up...

Russian uses:

• какие бы вопросы ни возникли

This is not really talking about past time. It is a standard grammatical pattern.

In standard Russian, after this бы ... ни pattern, the past tense form is the usual and expected choice:

• Какие бы вопросы ни возникли...

Using возникнут here would sound wrong or at least nonstandard in this structure.

So even if the meaning refers to the future, Russian still normally says:

• что бы ни случилось
• как бы он ни старался
• какие бы вопросы ни возникли

This is something learners usually need to memorize as a pattern.

На семинаре is prepositional case after на when it means at or during an event/location.

• семинар → на семинаре

Here it means:

• at the seminar
• or during the seminar

Compare:

• на столе = on the table
• на лекции = at the lecture
• на семинаре = at the seminar

So this phrase tells us where / in what setting the questions may arise.

Because the verb ответить / отвечать takes the preposition на when the object is a question, request, letter, criticism, etc.

So in Russian you say:

• ответить на вопрос = answer a question
• ответить на письмо = reply to a letter

Here них refers back to вопросы, so:

• ответит на них = will answer them

This is different from English, where answer usually takes a direct object with no preposition.

It is accusative plural.

The preposition на here goes with ответить на что?

• на вопросы
• на них

Since вопросы is plural and inanimate, the accusative plural looks the same as the nominative plural:

• вопросы
• них is the pronoun form used after a preposition

So:

• они = they
• их = them
• but after a preposition: на них, о них, для них

Russian, like English, often avoids repetition when the meaning is clear.

So instead of saying:

• преподаватель ответит на вопросы

after already mentioning вопросы, Russian naturally uses:

• преподаватель ответит на них

This sounds smoother and more natural. Repeating the noun would still be possible, but the pronoun is stylistically better here.

Без спешки means:

• without haste
• without rushing
• unhurriedly
• taking one’s time

It suggests that the teacher will answer calmly and carefully, not in a hurried way.

Structure:

• без
  • genitive
• спешка → без спешки

Similar expressions:

• без ошибок = without mistakes
• без проблем = without problems
• без подготовки = without preparation

Ответит is perfective future, from ответить. It presents the action as a completed result:

• преподаватель ответит на них = the teacher will answer them

This fits well because the sentence focuses on the fact that every question will receive an answer.

If you said будет отвечать, that would be imperfective future and would emphasize the process:

• will be answering

That is possible in some contexts, but here ответит is more natural because the point is the successful completion of answering the questions.

Yes, Russian word order is flexible, but the current order is very natural.

The sentence is structured like this:

1. subordinate clause: Какие бы вопросы ни возникли на семинаре
2. main clause: преподаватель ответит на них без спешки

This mirrors the logic:

• Whatever questions come up at the seminar, the teacher will answer them without rushing.

You could move some parts around for emphasis, for example:

• На какие бы вопросы ни возникли на семинаре... would be wrong
• Преподаватель без спешки ответит на них, какие бы вопросы ни возникли на семинаре is grammatically possible but much less natural

So the original order is preferred because it clearly sets up the whatever X happens clause first.

Yes, возникнуть is a very natural verb with вопрос.

It means:

• to arise
• to come up
• to appear

So:

• вопросы возникли = questions arose / came up

This is a standard collocation in Russian.

Other possibilities exist, but they give slightly different nuances:

• появиться = to appear
• задать = to ask, but that would shift the focus to the people asking
• обсуждаться = to be discussed

Here, возникли fits especially well because it focuses on questions that naturally come up during the seminar.

A very literal translation would sound slightly unnatural in English. Russian and English use different structures here.

Close literal version:

• Whatever questions should arise at the seminar, the teacher will answer them without haste.

More natural English versions:

• Whatever questions come up during the seminar, the teacher will answer them without rushing.
• No matter what questions arise at the seminar, the teacher will answer them calmly.

So the grammar should not always be translated word for word, especially with какие бы ... ни and без спешки.