Quand elle pleure, sa mère lui chante une berceuse et lui fait un câlin.

Yes. Quand can mean both when and whenever, depending on context.

In this sentence, Quand elle pleure can be understood as:

• When she cries
• Whenever she cries

Because the rest of the sentence describes what her mother does in response, English often naturally uses whenever to express a repeated action.

Elle pleure is the normal, simple way to say she cries or she is crying, depending on context.

French uses the present tense more broadly than English:

• elle pleure = she cries / she is crying

Être en train de + infinitive is possible, but it adds emphasis to the idea that the action is happening right now. In most everyday sentences, French just uses the plain present tense.

Yes — sa can mean his, her, or even its, depending on context.

French possessive adjectives agree with the thing possessed, not with the owner.

Here:

• mère is feminine singular
• so French uses sa

That means:

• sa mère = his mother or her mother

In this sentence, the context makes it clear that it means her mother.

Here, lui means to her.

It is an indirect object pronoun.

So:

• sa mère lui chante une berceuse = her mother sings a lullaby to her
• sa mère lui fait un câlin = her mother gives her a hug/cuddle

French often uses lui where English uses her after a verb plus to.

Because it belongs to both verbs.

The sentence has two actions:

• lui chante une berceuse
• lui fait un câlin

In French, object pronouns normally go directly before the verb they belong to. So when you have two verbs joined by et, you usually repeat the pronoun:

• sa mère lui chante une berceuse et lui fait un câlin

That is the normal and natural structure.

Because the person receiving the action is an indirect object, not a direct object.

Compare:

• chanter une berceuse à quelqu’un = to sing a lullaby to someone
• faire un câlin à quelqu’un = to give someone a hug/cuddle

That à quelqu’un becomes lui in pronoun form.

So:

• à elle → lui

You would use la for a feminine direct object, not for to her.

Because French usually requires an article where English sometimes does not.

So French says:

• chanter une berceuse = literally sing a lullaby
• faire un câlin = literally do/give a hug

Using une and un is completely normal here. French generally does not drop the article the way English sometimes can.

Literally, faire un câlin means something like to do a cuddle/hug, but that literal wording is not how you should think of it in English.

It is a very common French expression meaning:

• to give a hug
• to cuddle
• sometimes to give a cuddle

So lui fait un câlin means gives her a hug or cuddles her, depending on tone and context.

A berceuse is specifically a lullaby — a song sung to soothe a child and help them sleep or calm down.

A chanson is just a general song.

So une berceuse is more precise and fits the context of comforting a crying child.

Because French object pronouns usually come before the conjugated verb.

So:

• sa mère chante une berceuse à elle becomes sa mère lui chante une berceuse
• sa mère fait un câlin à elle becomes sa mère lui fait un câlin

This is standard French pronoun placement.

Because the sentence begins with a subordinate clause:

• Quand elle pleure, ...

When a time clause like this comes first, French often separates it from the main clause with a comma, just as English often does:

• When she cries, her mother...

It helps mark the pause and makes the structure clearer.

It can suggest a habitual or repeated action more naturally than a single one.

So the sentence often feels like:

• Whenever she cries, her mother sings her a lullaby and gives her a hug

French present tense is commonly used for general truths, habits, and repeated actions. Without more context, that is probably the most natural reading here.

Grammatically, à elle exists, but here lui is the normal choice.

• sa mère chante une berceuse à elle sounds heavy or emphatic
• sa mère lui chante une berceuse is the ordinary version

French usually prefers the pronoun lui unless you specifically want contrast or emphasis, such as:

• Sa mère lui chante une berceuse à elle, pas à son frère.

So in this sentence, lui is exactly what you would expect.