Какую бы книгу ни взяла моя дочь, она всё равно сначала смотрит на картинки.

Because книгу is the direct object of взяла.

• Dictionary form: книга
• Nominative question: какая книга?
• Accusative question/object form: какую книгу?

Since the daughter is taking a book, Russian uses the accusative case: взять какую книгу? → какую книгу.

Here бы helps create a free-choice / concessive meaning: whatever book, whichever book, no matter which book.

This is a very common Russian pattern:

• кто бы ни... = whoever
• что бы ни... = whatever
• где бы ни... = wherever
• какой бы ни... = whichever / no matter which

So какую бы книгу... does not mean simple past or conditional by itself here. It is part of a set expression meaning any book that she may take.

Because this construction uses ни, not не.

In Russian, the pattern question word + бы + ни + verb expresses no matter... / whoever... / whatever...:

• Что бы он ни сказал... = No matter what he says...
• Куда бы мы ни поехали... = Wherever we go...
• Какую бы книгу ни взяла... = Whichever book she takes...

If you used не instead, it would be ordinary negation, which is not the meaning here. So ни is required in this structure.

This is one of the trickier parts for learners.

After бы, Russian often uses the past-tense form of the verb, but the meaning is not necessarily past. In this pattern, the past form helps express a hypothetical, indefinite, or repeated possible situation.

So ни взяла here means something like:

• whichever book she takes
• whatever book she may pick up

It is not describing one completed past event.

Also, взяла is feminine singular because the subject is моя дочь.

Because Russian past-tense verbs agree with gender and number.

• взял = masculine
• взяла = feminine
• взяло = neuter
• взяли = plural

The subject is моя дочь, which is feminine, so the verb is взяла.

Russian often uses the generic singular in this kind of statement.

So какую бы книгу ни взяла means whichever book she takes, one book at a time, but as a general repeated idea. English often does the same:

• Whatever book she takes...
• Any book she picks up...

So singular книгу is completely natural here.

Because the sentence describes a habitual action: this is what she does in general.

Russian uses the present tense of an imperfective verb for repeated or typical behavior:

• она сначала смотрит на картинки = she first looks at the pictures

This means whenever this situation happens, that is her usual reaction. It is not about one single future event, so the present tense is natural.

Всё равно means something like:

• all the same
• anyway
• in any case

It emphasizes that the result does not change:

• regardless of which book she takes,
• she still looks at the pictures first.

So it strengthens the idea of no matter what, the same thing happens.

Because the verb смотреть normally takes на + accusative when it means to look at something.

So:

• смотреть на кого? на что?
• смотреть на картинки

This is just the normal government pattern of the verb.

Compare:

• видеть картинки = to see the pictures
• смотреть на картинки = to look at the pictures

Russian word order is flexible, and this order is quite natural after the fronted phrase какую бы книгу ни...

The sentence is built around the whole concessive clause:

• Какую бы книгу ни взяла моя дочь...

Placing взяла before моя дочь keeps the clause flowing smoothly and sounds idiomatic. You may also encounter other orders in Russian, but this one is very natural in writing and speech.

So this is less about strict grammar and more about style and information flow.

Yes — that is a very good way to learn it.

It is best to treat this as a common Russian pattern:

• кто бы ни... = whoever
• что бы ни... = whatever
• где бы ни... = wherever
• когда бы ни... = whenever
• какой бы ни... = whichever / no matter which

So in your sentence:

• какую = feminine accusative singular form of какой
• бы
• ни
• verb in past-form shape

Together they create the meaning no matter which book.

If you memorize the pattern as a chunk, it becomes much easier to recognize and use.