Какие бы книги ни лежали на столе, дочка всё равно берёт ту, в которой есть картинки.

This is a very common Russian pattern meaning whatever / no matter which.

• какие = which / what kind of
• бы adds a hypothetical or unrestricted sense
• ни helps create the meaning no matter which

So:

• Какие бы книги ни лежали на столе...
  = Whatever books are lying on the table...
  = No matter which books are on the table...

You can learn this as a set pattern:

• кто бы ни... = whoever
• что бы ни... = whatever
• где бы ни... = wherever
• какие бы ни... = whatever kind of / whichever

Because ни is the normal particle used in this concessive pattern.

Compare:

• не usually negates something directly
• ни is used in patterns like кто бы ни, что бы ни, как бы ни, какие бы ни

So in this sentence, ни does not simply mean plain negation. It helps build the meaning no matter what / whichever.

That is why:

• Какие бы книги не лежали... would sound wrong here
• Какие бы книги ни лежали... is the correct structure

бы in Russian is not only for if-type conditionals. It is also used in patterns expressing:

• hypothesis
• possibility
• concession
• lack of restriction

Here it helps create the idea of regardless of which books happen to be there.

So although English may not show anything like this very clearly, Russian uses бы as part of the fixed pattern:

• какие бы ... ни
• кто бы ... ни
• что бы ... ни

This is one of those structures you should recognize as a whole.

Because книги is the subject of лежали.

In the clause:

• книги лежали на столе = the books were lying on the table

the books are the things doing the lying, so they are in the nominative.

And какие agrees with книги:

• какие = nominative plural
• книги = nominative plural

So какие бы книги ни лежали... is grammatically parallel to какие книги лежат...

Not necessarily. This is a very important point.

In Russian, the form used with бы is historically the past tense form, but in many cases it does not actually mean past time. It functions as part of an irrealis / hypothetical construction.

So:

• лежали here is not really they lay in the past
• it is part of the structure какие бы ... ни лежали

The meaning is more like:

• whatever books may be lying there
• no matter which books happen to be lying there

So the clause refers to a general situation, not specifically to the past.

всё равно means something like:

• all the same
• anyway
• regardless
• still

It emphasizes that the result does not change.

So the logic is:

• no matter which books are on the table,
• the daughter still / anyway takes the one with pictures.

Without всё равно, the sentence would still make sense, but всё равно strengthens the idea that her choice is always the same.

Because ту is the feminine singular accusative form of та.

The verb берёт takes a direct object, and that object is understood as книгу:

• дочка берёт ту (книгу) = the daughter takes that one / that book

Forms of тот:

• masculine: тот
• feminine: та
• neuter: то
• plural: те

Accusative feminine singular:

• ту

So ту agrees with the implied noun книгу.

Because Russian often omits a noun when it is obvious from context.

Here, ту means:

• that one
• literally that [book]

Since the whole sentence is already about books, repeating книгу is unnecessary.

Both are possible:

• дочка всё равно берёт ту, в которой есть картинки
• дочка всё равно берёт ту книгу, в которой есть картинки

The version without книгу sounds natural and efficient.

Because this is a relative clause: the one in which there are pictures.

The implied noun is книга, which is:

• feminine
• singular

So the relative pronoun must agree with книга:

• nominative feminine singular: которая
• prepositional feminine singular after в: которой

Why prepositional? Because Russian says:

• в книге = in the book

So:

• в которой есть картинки = in which there are pictures

This is the same pattern as:

• дом, в котором я живу = the house in which I live
• город, в котором она родилась = the city in which she was born

Here есть means there are in an existential sense.

So:

• в которой есть картинки = in which there are pictures

Russian often uses есть when introducing the existence of something somewhere.

Compare:

• В книге есть картинки. = There are pictures in the book.
• В книге картинки красивые. = The pictures in the book are beautiful.

In your sentence, the idea is existence, not description, so есть fits naturally.

Yes.

• дочь = the basic word daughter
• дочка = a more affectionate, everyday, colloquial form

Both can mean daughter, but дочка often sounds warmer and more natural in ordinary speech.

So here it gives the sentence a slightly homier, more personal tone, as if talking about a child in a family setting.

Yes, to some extent. Russian word order is flexible, but changes in order can affect emphasis and style.

For example:

• Какие бы книги ни лежали на столе, дочка всё равно берёт ту, в которой есть картинки.

This version is very natural. It first sets up the regardless of circumstances idea, then gives the result.

You could also say:

• Дочка всё равно берёт ту, в которой есть картинки, какие бы книги ни лежали на столе.

This is possible, but less neutral and more stylistically marked.

So the original order is good because it clearly presents:

1. the irrelevant variable: whatever books are there
2. the consistent outcome: she picks the one with pictures

Most naturally, it describes a habitual tendency or a general truth.

That is because:

• берёт is imperfective present
• the whole sentence sounds like a general observation about what the girl typically does

So it means something like:

• whenever there are books on the table, she ends up choosing the pictured one

Russian present tense often works this way for repeated behavior, just like English:

• She always takes the one with pictures.

So even though it is present tense, it is not limited to one single moment.