Сколько бы раз малышка ни капризничала перед сном, папа всё равно разговаривает с ней мягко и спокойно.

This is a very common Russian concessive pattern. It means:

• no matter how many times ...
• however many times ...

So:

• Сколько бы раз малышка ни капризничала ... = No matter how many times the little girl fusses / acts up ...

How it is built:

• сколько = how many
• раз = times
• бы = adds a hypothetical / indefinite sense
• ни = helps create the meaning no matter ...

This pattern often pairs with всё равно in the main clause:

• Сколько бы раз ... ни ..., папа всё равно ...
• No matter how many times ..., dad still ...

After бы, Russian often uses the past tense form to express something that is not tied to real past time. It is part of an irrealis / subjunctive-like construction.

So here:

• капризничала does not literally mean she was acting up
• it is the form Russian normally uses after бы in this kind of pattern

Why капризничала specifically?

• the subject is малышка, which is feminine singular
• so the verb takes the feminine singular past form: капризничала

Compare:

• что бы он ни делал = no matter what he does
• где бы они ни жили = no matter where they live
• сколько бы раз она ни капризничала = no matter how many times she acts up

So the form looks past, but the meaning here is more like English does / may do.

Because this is not ordinary negation.

In constructions like:

• кто бы ни
• что бы ни
• где бы ни
• сколько бы ни
• как бы ни

Russian uses ни to mean something like:

• no matter who
• no matter what
• no matter where
• however much
• however

So in your sentence:

• Сколько бы раз ... ни капризничала = No matter how many times ... acts up

Here ни is part of the fixed concessive pattern. It does not simply mean not.

Using не here would sound wrong or would change the structure completely.

Сном is instrumental singular of сон.

It is used because of the preposition перед, which normally takes the instrumental case.

So:

• сон = sleep
• перед сном = before sleep / before bedtime

This is a very common expression:

• перед уроком = before class
• перед ужином = before dinner
• перед сном = before sleep / before bed

In English we usually say before bed or before bedtime, but Russian often uses перед сном literally before sleep.

Because the verb разговаривать normally goes with с + instrumental when it means to talk with / to speak with someone.

So:

• разговаривать с кем? = to talk with whom?
• с ней = with her

That is why the pronoun is in the instrumental form:

• она → ней after a preposition
• с ней = with her

Compare two different verbs:

• разговаривать с ней = to talk with her
• говорить ей = to say / tell to her, to speak to her

Both can sometimes translate as talk to her, but the grammar is different.

Всё равно means:

• all the same
• still
• anyway
• regardless

In this sentence it shows that the father’s behavior does not change, despite the child’s fussing.

So the structure is:

• No matter how many times she fusses before bed, dad still talks to her gently and calmly.

Without всё равно, the sentence would still be understandable, but всё равно makes the contrast clearer and more natural.

It is very common in Russian with this type of concessive clause.

Both verbs relate to speaking, but they are not exactly the same.

• говорить = to say, to speak
• разговаривать = to talk, to converse, to speak with someone

Here разговаривает с ней emphasizes the way the father interacts with her. It sounds natural when talking about how someone addresses another person.

So:

• папа разговаривает с ней мягко и спокойно = dad talks to her gently and calmly

If you used говорит с ней, it would sound less natural here. Russian usually prefers разговаривать с кем-то for talk with someone.

Because they describe how the father talks.

Russian uses adverbs here just like English often uses:

• gently
• calmly

So:

• мягкий = soft, gentle
• мягко = gently, softly

and

• спокойный = calm
• спокойно = calmly

They modify the verb разговаривает:

• разговаривает как?
• мягко и спокойно
• How does he talk?
• Gently and calmly

It sounds more like a general or repeated situation, not one single event.

Why?

• the main verb is разговаривает, present tense imperfective
• the whole sentence describes a typical pattern of behavior
• сколько бы раз ... ни ... naturally suggests repeated occurrences

So the sense is something like:

• Whenever this happens, he still speaks to her gently and calmly
• No matter how many times she acts up before bed, he still talks to her gently and calmly

If Russian wanted to focus on one specific past occasion, the sentence would probably be built differently.

Малышка is an affectionate word. It can mean:

• little girl
• little one
• baby or small child in some contexts

It is warmer and more tender than a neutral word like ребёнок.

So in this sentence, малышка helps create a gentle, family tone. It sounds like the speaker is referring to the child affectionately, not just neutrally describing her.